xr-ee-eme_2006_004
TRANSCRIPT
EVALUATION OF MOTOR ONLINE DIAGNOSIS BY FEM SIMULATIONS
Thanis Sribovornmongkol
Master’s Thesis XR-EE-EME 2006:04
Electrical Machines and Power Electronics School of Electrical Engineering
Royal Institute of Technology Stockholm 2006
I
ABSTRACT
Early detection of abnormal conditions during motors operation would eliminate consequential damages on motors, so outage time and costs of repairing can be reduced. Due to unique fingerprints from faults in line currents, it is possible to detect faults by extracting fault information from line currents, which is so-called, Motor Current Signature Analysis.
In this thesis, Finite Element Method has been implemented to simulate three main kinds of faults; rotor asymmetry, airgap asymmetry and stator asymmetry. Time-Stepping FEM simulation models have been developed for induction motors, and the various faults have been implemented to the models. Then, three different approaches; i.e. Motor Current Signature Analysis, Instantaneous Power Approach and The Extended Park’s Vector, based on simple measurement have been applied to extract fault information from the FEM simulation results, and the evaluation of three approaches has been presented.
Besides, two cases of operating conditions, which are unsymmetrical supplied voltages and oscillating loads, have been described. In addition, the evaluated approach has been applied to real measurement. The possible root causes of the inexplicable components in the real measurement have been described.
Keywords: Condition Based Monitoring, Online Monitoring, Fault Diagnosis, Finite Element Method, Motor Current Signature Analysis, Instantaneous Power Approach, The Extended Park’s Vector, Frequency Analysis, Rotor Asymmetry, Airgap Asymmetry, Stator Asymmetry
II
SUMMARY
In this study, FEM simulations have been implemented to study fault phenomena in induction motors. Unique signatures in the electrical supply measurements are identified for Online Motor Diagnosis. The four main ideas presented in this study are;
• Theoretical derivations of harmonics in the stator currents generated by the most common faults
• Implementation of FEM simulation for fault studies
• Evaluation of three different approaches for Online Motor Diagnosis by simple measurement on the supply side
• Application in a case study with the aim to find possible root causes for abnormal harmonic sidebands in the stator currents
The six most common faults are Rotor broken bars, Rotor broken end rings, Static eccentricity, Dynamic eccentricity, Mix of static and dynamic eccentricities and Inter-turn short circuit. In most cases, each has a unique fault signature; however, there is a risk of confusion in some cases as presented in the report.
The various faults have been studied both individually and in combination. The simulation results agree well with the theory, but some unexpected harmonics are still present. The conclusion is that FEM simulation is a powerful tool to study fault characteristics, but it is time-consuming.
Three analysis approaches, Motor Current Signature Analysis (MCSA), Instantaneous Power Approach (IPA) and The Extended Park’s Vectors (EPVA), have been applied to the simulation results and are evaluated. The conclusion is that EPVA is recommended if reference measurements are available, otherwise IPA is preferred.
Finally, a set of real measurement data is analyzed, resulting in the conclusion that the most likely root cause for the inexplicable sidebands is the effect from load oscillations since no other fault can generate such harmonics.
III
ACKNOWLEDGEMENTS
This master thesis has been carried out at Department of Electrical Machine and Intelligent Motion, ABB Corporate Research, Västerås, Sweden, during September2005-April2006.
Firstly, I would like to express my heartfelt gratitude to my supervisor, Christer Danielsson for his guidance and encouragement, which he gave me throughout my works. Without his advices, I believe that it would have been impossible to finish this thesis.
I would also like to show my deep gratitude to my supervisor and examiner from KTH, Prof. Chandur Sadarangani, for his advices and supports. I always felt enthusiastic after having a conversation with him.
Many thanks also go to Sture Erixon for his supports on information for this project. My special thanks would be presented to Robert J Anderson for his assistance on FLUX2D, which is one of the most difficult software I have ever used before.
I am obliged to Dr. Heinz Lendenmann, who is the Manager at Department of Electrical Machine and Intelligent Motion, for giving me a chance to do my thesis at ABB Corporate Research.
Lastly, without them, I would have had today. I would like to present my loves and profound gratitude to my parents, Dad & Mom. Thank you for their moral supports, cares and encouragement.
Thanis Sribovornmongkol
Västerås, Sweden
April 2006
“There is nothing that perseverance cannot Win”
IV
TABLE OF CONTENTS
1 INTRODUCTION.............................................................................................................................. 1
2 TIME HARMONICS AND SPACE HARMONICS IN INDUCTION MOTORS ................................. 3 2.1 INTRODUCTION............................................................................................................................ 3 2.2 TIME HARMONICS AND SPACE HARMONICS................................................................................... 5 2.3 TIME HARMONICS ON AIRGAP MMF FROM TIME HARMONICS IN SUPPLY VOLTAGES........................ 5 2.4 MMF SPACE HARMONICS............................................................................................................ 6
2.4.1 Stator MMF ........................................................................................................................ 6 2.4.2 Rotor MMF......................................................................................................................... 7 2.4.3 Total MMF.......................................................................................................................... 8
2.5 AIRGAP PERMEANCE ................................................................................................................... 8 2.6 AIRGAP FLUX DENSITY ................................................................................................................ 9
2.6.1 Stator Flux Density ............................................................................................................ 9 2.6.2 Rotor Flux Density ........................................................................................................... 11
2.7 CONCLUSION ............................................................................................................................ 11 3 MOTOR CURRENT SIGNATURE ANALYSIS (MCSA)................................................................ 12
3.1 INTRODUCTION.......................................................................................................................... 12 3.2 ROTOR ASYMMETRY.................................................................................................................. 12
3.2.1 Cause for Rotor Breakage ............................................................................................... 12 3.2.2 Detection of Rotor Breakage ........................................................................................... 13
3.3 AIRGAP ASYMMETRY ................................................................................................................. 16 3.3.1 Static eccentricity............................................................................................................. 17 3.3.2 Dynamic eccentricity........................................................................................................ 18 3.3.3 Mix eccentricity ................................................................................................................ 20
3.4 STATOR ASYMMETRY ................................................................................................................ 21 3.4.1 Cause for Inter-Turn Short Circuit.................................................................................... 21 3.4.2 Detection of Inter-Turn Stator Winding Fault ................................................................... 22
3.5 BEARING DAMAGES ................................................................................................................... 23 3.5.1 Cause for Bearing Damages ........................................................................................... 23 3.5.2 Detection of Bearing Damages........................................................................................ 24
3.6 MECHANICAL COUPLING ............................................................................................................ 25 3.7 OSCILLATING LOADS.................................................................................................................. 25 3.8 CONCLUSION ............................................................................................................................ 26
4 INSTANTANEOUS POWER APPROACH & THE EXTENDED PARK’S VECTOR APPROACH27 4.1 INTRODUCTION.......................................................................................................................... 27 4.2 INSTANTANEOUS POWER APPROACH.......................................................................................... 27 4.3 THE EXTENDED PARK’S VECTOR APPROACH .............................................................................. 29 4.4 CONCLUSION ............................................................................................................................ 31
5 FEM SIMULATION MODELS........................................................................................................ 32 5.1 INTRODUCTION.......................................................................................................................... 32 5.2 FINITE ELEMENT MODELING........................................................................................................ 32 5.3 PHYSICAL MODEL OF INDUCTION MOTORS.................................................................................. 32 5.4 ROTOR ASYMMETRY.................................................................................................................. 36
5.4.1 Broken Rotor Bar ............................................................................................................. 36 5.4.2 Broken Rotor End Ring.................................................................................................... 38
5.5 AIRGAP ASYMMETRY ................................................................................................................. 40 5.5.1 Static Eccentricity ............................................................................................................ 40 5.5.2 Dynamic Eccentricity ....................................................................................................... 43 5.5.3 Mix of Static and Dynamic Eccentricities......................................................................... 44
5.6 STATOR ASYMMETRY ................................................................................................................ 48 5.6.1 Inter-turn Short Circuit on Stator Windings...................................................................... 48
5.7 CONCLUSION ............................................................................................................................ 53 6 FEM SIMULATION – MOTOR CURRENT SIGNATURE ANALYSIS .......................................... 54
V
6.1 INTRODUCTION.......................................................................................................................... 54 6.2 FEM MODEL.............................................................................................................................. 54 6.3 SINGLE FAULT ........................................................................................................................... 55 6.4 FAULT COMBINATIONS ............................................................................................................... 61
6.4.1 Combination of Rotor and Airgap Asymmetry ................................................................. 61 6.4.2 Combination of Rotor and Stator Asymmetry .................................................................. 64 6.4.3 Combination of Airgap and Stator Asymmetry ................................................................ 65 6.4.4 Combination of Rotor, Airgap and Stator Asymmetry ..................................................... 68
6.5 UNSYMMETRICAL SUPPLY VOLTAGE & OSCILLATING LOADS ........................................................ 69 6.6 CONCLUSION ............................................................................................................................ 71
7 FEM SIMULATION (CONTINUED) – INSTANTANEOUS POWER APPROACH........................ 73 7.1 INTRODUCTION.......................................................................................................................... 73 7.2 SINGLE FAULT ........................................................................................................................... 73 7.3 FAULT COMBINATIONS ............................................................................................................... 77 7.4 UNSYMMETRICAL SUPPLY VOLTAGE & LOAD OSCILLATION .......................................................... 84 7.5 CONCLUSION ............................................................................................................................ 85
8 FEM SIMULATION (CONTINUED) – THE EXTENDED PARK’S VECTOR APPROACH........... 87 8.1 INTRODUCTION.......................................................................................................................... 87 8.2 SINGLE FAULT ........................................................................................................................... 87 8.3 FAULT COMBINATIONS ............................................................................................................... 93
8.3.1 Combination of Rotor and Airgap Asymmetry ................................................................. 93 8.3.2 Combination of Rotor and Stator Asymmetry .................................................................. 96 8.3.3 Combination of Airgap and Stator Asymmetry ................................................................ 97 8.3.4 Combination of Rotor, Airgap and Stator asymmetry.................................................... 101
8.4 UNSYMMETRICAL SUPPLY VOLTAGE & LOAD OSCILLATION ........................................................ 102 8.5 CONCLUSION .......................................................................................................................... 103 8.6 EVALUATION FOR MCSA, IPA AND EPVA ................................................................................ 104
9 IMPLEMENTATION..................................................................................................................... 106 9.1 INTRODUCTION........................................................................................................................ 106 9.2 RAW MEASUREMENT DATA ...................................................................................................... 106 9.3 SIMPLE INVESTIGATION............................................................................................................ 107 9.4 DATA PREPARATION ................................................................................................................ 108
9.4.1 Number of Sample......................................................................................................... 108 9.5 INVESTIGATION........................................................................................................................ 108
10 CONCLUSION AND FUTURE WORKS .................................................................................. 115 10.1 CONCLUSION....................................................................................................................... 115 10.2 FUTURE WORKS .................................................................................................................. 116
11 REFERENCES ......................................................................................................................... 117
LIST OF SYMBOLS............................................................................................................................ 122
APPENDIX .......................................................................................................................................... 123 I. COMPARISON OF THE AIRGAP LENGTH FOR VARIOUS TYPES OF ECCENTRICITIES. ................................ 123 II. MATLAB SCRIPT FOR THE AIRGAP CALCULATION.............................................................................. 124 III. FEM SIMULATION.......................................................................................................................... 125 IV. DYNAMIC MODEL OF AN INDUCTION MOTOR ................................................................................... 145
1
1 INTRODUCTION Robust induction motors are the most widely used electrical machines in industry. The unexpected down time of induction motors can cause production and revenue losses. It is, therefore, important to prevent unscheduled downtime, which can help not only to reduce maintenance costs but also to gain up income of industry.
In the survey report by EPRI [58], it presented the percentage failure for the wide range of induction motors. As can be seen in Figure 1.1, the survey found that 37% of motor failures were caused by stator winding failures, 10% by rotor failures, 41% by bearing failures and 12% by miscellaneous failures.
This master thesis was initiated from an inexplicable measurement of one induction motor. That initiated an idea to implement FEM simulations to study fault phenomena in induction motors, so failures could be identified before machines would be dismantled. This can enable the application on Online Motor Diagnosis.
The purposes of the thesis are to study influences from the most common faults on electrical measurement based on the stator side by implementing FEM simulations, and to apply three different approaches, which are Motor Current Signature Analysis (MCSA), Instantaneous Power Approach (IPA) and The Extended Park’s Vector (EPVA), to identify faults. Besides, the other purpose is to find the possible root causes for the inexplicable sidebands found in the measurement.
This thesis is composed of three main parts. The first part consisting of Chapter 2, 3 and 4 discusses knowledge in diagnostic fields. The theoretical studies on influences from faults on stator currents are introduced. Chapter 5, 6, 7 and 8 are the second part presenting Implementation of FEM simulations on fault studies and implementation of three diagnostic methods on simulation results. Then the inexplicable measurement has been investigated in Chapter 9. The objective in each chapter is generally presented below.
Chapter 2 provides general knowledge of time and space harmonics in induction motors. This is the main background for understanding influences from faults presented in Chapter 3.
Chapter 3 introduces causes of faults and influences on stator currents. Fault signatures in stator currents called Motor Current Signature Analysis (MCSA) are presented. The background from Chapter 2 is used to derive analytical expressions for fault indicators in stator currents.
Chapter 4 introduces the other two approaches that are Instantaneous Power Approach (IPA) and The Extended Park’s Vector (EPVA).
Figure 1.1 Percentage failure by component [58]
2
Chapter 5 presents FEM simulation models. The knowledge on fault characteristics from Chapter 3 is applied to build up FEM simulation models.
Chapter 6, 7 and 8 present FEM simulation results according to Chapter 5. The implementations of three different approaches presented in Chapter 3 and 4 are discussed.
Chapter 9 discusses the inexplicable measurement. The knowledge gained from the previous chapters is applied to investigate the measurement.
Chapter 10 presents the conclusion and future works.
Figure 1.2 Thesis Structure
3
2 TIME HARMONICS AND SPACE HARMONICS IN INDUCTION MOTORS
2.1 Introduction
Normally, in the study of induction machines, some simplified assumptions are made. For example, purely sinusoidal supply voltages, uniform airgap, infinite numbers of rotor and stator slots etc. These simplified assumptions are useful when the principle operation of induction machines is studied. However, they are not applicable for diagnostic problems since unsymmetrical conditions must be considered.
This chapter introduces time and space harmonics in the airgap flux density, which are influenced from non-sinusoidal quantities. The approach is to analyze the airgap flux density using analytical expression for the airgap MMF and permeance. Figure 2.1 summarizes the general process of harmonic generation in the airgap of induction motors.
This chapter is composed of five main sections. The purpose of the first section in 2.2 is to present how different time and space harmonics are. The characteristics of time and space harmonics are discussed. The second in 2.3 explains influences on the airgap MMF from time harmonics in supply voltages. Then, the section 2.4 discusses the airgap MMF space harmonics influenced from stator and rotor currents. Next, the expression of the airgap permeance influenced from non-constant airgap length is presented in the section 2.5. The description and expression in the section 2.4 and 2.5 will be applied to determine the airgap flux density in the section 2.6. The frequencies of induced stator and rotor quantities are discussed. The conclusion is presented in the section 2.7.
4
Figure 2.1 Schematic diagram of harmonics in induction machines
5
2.2 Time Harmonics and Space Harmonics
Any quantity that changes when time is varied is a time dependent quantity; on the other hand, any quantity that distributes in distance is a space dependent quantity. In Figure 2.2, time and space dependent waveform are shown.
According to Fourier analysis, any waveform or dependent quantity can be represented by a series of sinusoidal terms. The fundamental frequency is called for the base frequency of the waveform. The terms, of which the frequency is multiples of the fundamental frequency, are called as harmonics.
(a) Space Harmonic (b) Time Harmonic
Figure 2.2 Time Dependent and Space Dependent Waveform
At a specific position in space, space harmonics pulsate at the fundamental frequency ω but move with the angular speed kω . On the other hand, time harmonics pulsate at the frequency ωk and move with the angular speed ωk , where k is a harmonic order.
In induction machines, space harmonic quantities can exist both in the airgap MMF due to winding distributions in slots and in permeance waves due to non-constant airgap length. Besides, time harmonic quantities can take place in the airgap MMF due to time harmonics in the supply voltages.
The approach to analyze the airgap flux density is to determine the airgap MMF and permeance functions. The airgap flux density can be expressed as below.
0 01( , ) ( , ) ( , )
( , ) ( , ) ( , ) ( , )
AB t F t F tt A tt F t
μ μθ θ θδ θ δ θθ θ
= ⋅ ⋅ = ⋅
= Λ ⋅
(2. 1)
Where, ),( tθΛ is an airgap permeance per unit area, ),( tF θ is airgap MMF and ),( tθδ is airgap length.
2.3 Time Harmonics on Airgap MMF from Time Harmonics in Supply Voltages
Purely sinusoidal supply voltages feeding to an induction motor can create purely sinusoidal currents in windings, so the fundamental airgap MMF arises. If the supply voltages are not purely sinusoidal, such as when a motor is fed from an inverter, currents are also not purely sinusoidal, but contain some time harmonics. Time harmonics in the airgap MMF can be
6
expressed as shown in Equation 2.2. The stator MMF generated from the magnetizing currents in each phase winding can be obtained. As the phase windings are placed at 2 3π from each other, the space shift will also contribute to the stator MMF [3].
)3
4cos(
)3
2cos(
)cos(
1,,
1,,
1,,
πφω
πφω
φω
−−=
−−=
−=
tkII
tkII
tkII
kmkc
kmkb
kmka
)3
4(cos)3
4cos(2
)3
2(cos)3
2cos(2
)cos()cos(2
1,,
1,,
1,,
πθπω
πθπω
θω
−−=
−−=
=
ptkIN
m
ptkIN
m
ptkIN
m
kmse
kc
kmse
kb
kmse
ka
(2. 2)
Therefore; )cos(22
31
,,,,. tkp
INmmmm kmse
kckbkakm ωθ −=++= (2. 3)
Where, k is a time harmonic order and p is a number of pole pairs.
From Equation 2.3, it can be seen that time harmonics in the stator MMF can be obtained from time harmonics in the supply voltages. In addition, the space shift of phase windings can contribute to space harmonics in the stator MMF as well.
2.4 MMF Space Harmonics
The airgap MMF is generated from stator and rotor currents. Due to the distribution of stator and rotor windings, space harmonics in the airgap MMF can take place. The total airgap MMF can be expressed by the sum of the stator and rotor MMF as below.
( , ) ( , ) ( , )s rF t F t F tθ θ θ= + (2. 4)
Consider a three-phase induction motor with 2 p poles, q slots per pole per phase, with phase-belts displaced in space by 2 3π radians around the airgap. Harmonics in the airgap MMF due to stator and rotor slots are considered [1,2].
2.4.1 Stator MMF
, 1 11
ˆ( , ) cos( )s s vv
F t F vp tθ θ ω ϕ∞
=
= − −∑ (2. 5)
Where, )16( 1 += gv with 1g = ,...3,2,1,0 ±±±
1,
23ˆ Ikkv
qNF dvpvvs π=
Where, 1ω is a fundamental angular frequency, 1ϕ is a fundamental phase angle, N is a number of conductors per slot, 1I is stator phase currents, pvk is a pitch factor and dvk is a winding distribution factor, and v is a space harmonic order.
From Equation 2.5, a mechanical angular speed of the v -th space harmonic is given by;
pgvpv )16( 1
11
+==
ωωω (2. 6)
A sign of the mechanical angular speed of the v -th space harmonic expresses the direction of the space harmonic wave. The positive sign; i.e. 1,7,13,...,v = means the harmonic wave
7
rotates as the same direction as the fundamental wave, but the negative sign, i.e. -5, -11, -17,...,v = means the harmonic wave rotates in the backward direction. In addition, stator slot harmonics and phase-belt space harmonics have to take into consideration due to their significance. For the stator slot harmonics, they can be found at the v -th harmonic order as below.
1( 1)sQv gp
= + or )16( 1 +sqg (2. 7)
Where, sQ is a number of stator slots, and sq is a number of stator slots per pole per phase. For the phase-belt harmonics, they are the harmonics, 5,7,11,13,...v = to the first stator slot harmonics.
2.4.2 Rotor MMF
The rotor MMF can be considered from two different sources, which are due to the fundamental rotor current and rotor harmonic currents.
2.4.2.1 Rotor MMF due to the Fundamental Rotor Current
,1 ,1 1 ,11
ˆ( , ) cos( )r rF t F p tμ μμ
θ μ θ ω ϕ∞
=
= − −∑ (2. 8)
Where, 1 2( 1)rQg
pμ = + With 2g = ,...3,2,1,0 ±±±
,1 1 1 13 2ˆ ( 1) cosr
qNF Iμμξ ξ ϕ
πμ= − −
Where, rQ is a number of rotor slots, μ is a space harmonic order, 1ϕ is a phase angle of stator currents, μϕ is a phase angle of rotor MMF harmonics, the parameters 1ξ and μξ are winding factors for the fundamental component and theμ -th harmonics respectively.
Hence, the angular speed of the airgap flux density generated by the rotor MMF due to the fundamental rotor current in the stator reference frame can be calculated as;
1 2
,1 1 11
(1 )( ) 1
r rs g Q spp p pμω ωω μ ω
μ⎡ ⎤−
= + = +⎢ ⎥⎣ ⎦
(2. 9)
Where, rω is an electrical rotor angular frequency, and s is slip.
2.4.2.2 Rotor MMF due to Rotor Current Harmonics
,2 ,2, 2 ,21
ˆ( , ) cos( )r rF t F p tμ μ μμ
θ μ θ ω ϕ∞
=
= − −∑ (2. 10)
Where, 2 2( )rQg vp
μ = + With 2g = ,...3,2,1,0 ±±±
,2,ˆ ( 1) 2r rv v
vF Iμμ ξ
μ= − −
8
Where, rvI is rotor end-ring harmonic current and vξ is a winding factor. The speed of the airgap flux density generated by rotor harmonic currents can be analyzed from harmonics of the stator flux density. The harmonics of the stator flux density rotate at the speed of 1 vpω and induce currents in the rotor windings with the frequency as;
12 1 1( ) (1 )r vp v s
vp pω ωω ω ω= − = − − (2. 11)
Thus, the speed of the airgap flux density generated by rotor harmonic currents relative to the stator frame can be calculated as;
1 1 2,2 2 1
2
(1 ) (1 )( ) 1r rv s g Q spp p pμω ω ωω μ ω
μ⎡ ⎤− − −
= + = +⎢ ⎥⎣ ⎦
(2. 12)
2.4.3 Total MMF
From Equation 2.5 to 2.11, the total MMF in Equation 2.4 can be expressed as below.
1 1
, 1 1 ,1 1
0 1 1
1 1 1 1 2 1 1 10,1,2,... 1, 2,...
3
ˆ ˆ( , ) cos( ) cos( )
ˆ cos( )ˆ ˆ cos((6 1) ) cos((6 1) )
ˆ cos((
s v rv
g g
s
F t F vp t F p t
F p t
F g p t F g p t
F Q
μ μ μμ
θ θ ω ϕ μ θ ω ϕ
θ ω ϕ
θ ω ϕ θ ω ϕ
∞ ∞
= =
= =− −
= ± − + ± −
= − −
+ + − − + + − −
+
∑ ∑
1 1 4 1 1
5 1 5 1
ˆ) ) cos(( ) ) cos(( )( ) ) cos(( )( ) )
s
r m r m
p t F Q p tF Q p t s t F Q p t s t
θ ω ϕ θ ω ϕθ ω ω θ ω ω
− − − + + − −+ − − + + + − −
(2. 13)
It should be noted that the stator MMF harmonics are influences from stator windings and slots, but the rotor MMF harmonics are influences from only rotor slots. Equation 2.13 can be simplified if only the most significant harmonics, such as the fundamental, the first order slot harmonics, phase-belt harmonics and winding harmonics, are considered.
2.5 Airgap Permeance
To determine an airgap permeance, the airgap length has to be considered. At a specific location in space, when a rotor is rotating, the airgap length is not constant, but it is changing. Consider the airgap shown in Figure 2.3, the function of the airgap length can be determined as shown in Equation 2.14. [1] 021 ),()(),( δθδθδθδ −+= tt (2. 14)
The airgap length can be separated into three parts, which are the constant airgap length 0δ , the space dependent airgap length 1δ and the time and space dependent airgap length 2δ . Therefore, the airgap permeance can be expressed as shown in Equation 2.15.
Figure 2.3 Airgap with both stator slots and rotor slot
9
[ ]
0
1 2 0
1 2 0
01, 2,
1 10 1 2
1,2,1 1
0 1 2 1,2
1( , ) 1 1 1( ) ( , )
1 cos( ) cos ( )
cos ( )
k s m r mk mc c
k m s r r mk m
tt
kQ mQ tk k
kQ mQ mQ t
μθδ θ δ θ δ
μ θ θ ωδ
θ ω
∞ ∞
= =
∞ ∞
±= =
Λ = =+ − + −
Λ Λ Λ
= + Λ + Λ −
⎡ ⎤+ Λ ± −⎣ ⎦
′ ′ ′= Λ +Λ +Λ +Λ
∑ ∑
∑∑
(2. 15)
Where, mω is mechanical rotational speed ( 1(1 )m r p s pω ω ω= = − )
It can be seen from Equation 2.15 that the airgap permeance is composed of four terms. The first term is the constant term that results from the constant airgap length. The second term represents the influences from stator slots. The third term can describe the effect from rotor slots related to numbers of rotor slots and rotor speed. Lastly, the influences from both stator and rotor slots are presented in the terms of sum and difference between numbers of stator and rotor slots.
2.6 Airgap Flux Density
The airgap flux density can be obtained by substituting Equation 2.13 and 2.15 into Equation 2.1, which gives the results in infinitely many terms. Nevertheless, the airgap flux density can be approximated by taking only the simplified MMF and the first three terms in the permeance expression into account as shown in Equation 2.16. The results are summarized as in Table 2.1. More details can be found in [1].
[ ]{[ ] [ ]
0 1 2 1,2 ,1 , ,1 ,
0 1 2
0 1 1 1
2 1 3 1
( , ) ( , ) ( , ) ( ) ( )
( cos cos ( ))
cos( ) cos 5
cos 7 cos ( )
s s v r r
s r r
s
B t t F tF F F F
b b Q b Q w t
a p t a p t
a p t a Q p t
μ
θ θ θ
θ θ
θ ω θ ω
θ ω θ ω
= Λ ⋅′ ′ ′= Λ +Λ +Λ +Λ ⋅ + + +
= + + −
− + + +
− + − + +
[ ] [ ][ ]}
4 1 5 1
6 1
cos ( ) cos ( )( )
cos ( )( )s r m
r m
a Q p t a Q p t s t
a Q p t s t
θ ω θ ω ω
θ ω ω
+ − + − − + +
+ − −
(2. 16)
2.6.1 Stator Flux Density
The influence from the airgap flux density generated from the stator currents is to induce voltages in the rotor windings. The expressions for the frequency of induced voltages can be determined as below;
2.6.1.1 Fundamental Stator Flux Density
,1 ,1 1 1( , ) cos( )s sB t B p tθ θ ω ϕ= − −) (2. 17)
Due to the difference between the rotational angular speed r pω and the synchronous speed of the fundamental airgap flux density 1 pω , voltages and currents can be induced in the rotor windings. The frequency of the induced rotor voltages and currents can be
10
determined from the relative angular speed between the rotational speed and the synchronous speed as shown in Equation 2.18.
12 1
1 ( )2
rf p sfp pω ω
π= − ⋅ = (2. 18)
2.6.1.2 Harmonics of Stator Flux Density
, 1 1ˆ( , ) cos( )sw s vB t B vp tθ θ ω ϕ= − − (2. 19)
Harmonics of the airgap flux density rotate at an angular speed of 1 vpω and pulsate at the fundamental frequency 1ω . This means the speed of harmonics decreases vp times of the fundamental component, but the pulsation is still constant, or the number of pulsations is vp times of the fundamental component. Therefore, the frequency of induced rotor voltages and currents from harmonics of the stator flux density is determined as shown in Equation 2.20.
[ ]12, 1
1 ( ) (1 ) 12
rvf vp s v f
p vpω ω
π= − ⋅ = − − (2. 20)
Forward Harmonic order
Frequency of induced stator
voltage and currentω
1 0 0 1cos( )a b p tθ ω− 1 1ω
2 ( )1 1
1cos2 s
a b Q p tθ ω⎡ ⎤+ −⎣ ⎦ 1sQp+ 1ω
3 [ ]2 0 1cos 7a b p tθ ω− 7 1ω
4 ( )4 0 1cos sa b Q p tθ ω⎡ ⎤+ −⎣ ⎦ 1sQ
p+ 1ω
5 ( )6 0 1cos ( )r ma b Q p t s tθ ω ω⎡ ⎤+ − −⎣ ⎦ 1rQ
p+ 1( 1)r
rQ sp
ω ω+ +
6 ( )4 0 1cos cos ( )s r ra b Q p t Q tθ ω θ ω⎡ ⎤+ − −⎣ ⎦
7 ( )6 1 1cos ( ) cosr r r sa b Q p Q t Q tθ ω ω⎡ ⎤+ − +⎣ ⎦ Backward 1 ( )1 1
1cos2 s
a b Q p tθ ω⎡ ⎤− +⎣ ⎦ 1sQp−
1ω
2 [ ]1 0 1cos 5a b p tθ ω+ 5 1ω
3 ( )3 0 1cos sa b Q p tθ ω⎡ ⎤− +⎣ ⎦ 1sQ
p− 1ω
4 ( )5 0 1cos ( )r ma b Q p t s tθ ω ω⎡ ⎤− − +⎣ ⎦ 1rQp− 1( 1)r
rQ sp
ω ω− −
5 ( )3 2 1cos cos ( )s r ra b Q p t Q tθ ω θ ω⎡ ⎤− + −⎣ ⎦
6 ( )5 1 1cos ( ) cosr r sa b Q p t s t Q tθ ω ω⎡ ⎤− − +⎣ ⎦
Table 2.1 Airgap flux density considering a finite number of space harmonics
11
2.6.2 Rotor Flux Density
Similar to the stator flux density, the flux density created by the rotor currents also induce voltages in the stator windings. The expression for the frequency of induced voltages can be determined as below;
2.6.2.1 Fundamental Rotor Flux Density
,1 ,1 1ˆ( , ) cos( )r rB t B p t μθ θ ω ϕ= − − (2. 21)
Due to the space-fixed stator windings, the rotor flux density also induces voltages in the stator windings as;
1, 1( 0)s induced p
pωω ω= − = (2. 22)
2.6.2.2 Harmonics of Rotor Flux Density
, ,ˆ( , ) cos( )r rB t B p tμ μ μ μθ μ θ ω ϕ= − − (2. 23)
Similar to the fundamental rotor flux density, harmonics of the rotor flux density rotate with the speed μω in the airgap. Thus, the frequency of induced voltages is μω .
2.7 Conclusion
In this chapter, the effect of time and space harmonics in the airgap flux density on the stator currents is discussed. Due to finite numbers of stator and rotor slots as well as distribution of stator windings, a non-sinusoidal airgap flux density will arise in the airgap. The influences from space harmonics of the airgap flux density can cause time harmonics in the supply voltages and currents, which cause pulsating torque. By determining the airgap MMF generated from stator and rotor currents and airgap permeance, the frequency of induced quantities in the rotor, airgap and stator, such as rotor voltages and currents, airgap flux density and stator voltages and currents can be determined. The details presented here are the main background for determining influences from faults on stator currents in the next chapter.
12
3 MOTOR CURRENT SIGNATURE ANALYSIS (MCSA)
3.1 Introduction
In this chapter, a technique to detect faults in induction machines by frequency analysis of stator currents is presented. Thanks to a unique consequence of each fault, unique harmonics in line currents are produced. By locating the specific harmonic components, which are called as Motor Current Signature Analysis (MCSA), faults in induction motors can be detected.
In this chapter, four fault types are presented. First, rotor asymmetry, which is the consequence of breakage in rotor bars or end rings, is introduced. Next, airgap asymmetry that is static, dynamic and mix of static and dynamic eccentricities is discussed. Then, stator asymmetry that is an inter-turn short circuit fault is described, and bearing faults are also reviewed. Moreover, influences from mechanical couplings in line currents are introduced. In addition, two operating conditions; unsymmetrical supply voltages and oscillating loads, of which consequences are similar to that of some faults, are presented.
The aim of this chapter is to study influences in stator currents from various faults. Causes, phenomena and characteristics of each fault type are presented. These will be applied to build FEM simulation models presented in Chapter 5. The background from the previous chapter is used to determine analytical expressions of fault indicators.
3.2 Rotor Asymmetry
According to the failure survey [58], it stated that about 10% of total failure cases related to rotor failures. One of rotor failures found frequently is rotor breakages. Breakages in a rotor can take place in rotor bars or rotor end rings. A broken piece of rotor bars or end rings can move along the airgap between the stator and rotor, and it can disrupt surfaces of stator windings leading to a sudden failure. This can result in high repairing costs and outage time. For this reason, the detection of rotor breakages at an early state is advantageous.
3.2.1 Cause for Rotor Breakage
Rotor breakages can be caused by many reasons. However, they can be summarized as;
1. Thermal stresses due to thermal overload and unbalance, hot spots or excessive losses, sparking (mainly fabricated rotor type)
2. Magnetic stresses caused by electromagnetic forces, unbalanced magnetic pull, electromagnetic noise and vibration
3. Residual stresses due to manufacturing problems
4. Dynamic stresses arising from shaft torque, centrifugal forces and cyclic stresses
5. Environmental stresses caused by contamination and abrasion of rotor material due to chemicals or moisture
6. Mechanical stresses due to loose laminations, fatigued parts, bearing failures and etc
7. Operating condition; pulsating load leading to rapid changes on the shaft torque
13
Risks of rotor failure can be reduced if these stresses can be kept under control. Designing, Building and Installation as well as Maintaining should be considered and done properly in order to reduce the stresses in a motor.
3.2.2 Detection of Rotor Breakage
There are many techniques to detect rotor breakages. In addition to the stator current method, negative sequence impedance or negative sequence current[35], zero sequence current[37], axial flux[36,40], torque [23], instantaneous power[21], the extend park’s vectors [24,25] , injection of low frequency signal [39] or vibration[36], are able to detect rotor breakages. However, determination of rotor breakages from stator currents is emphasized in this study.
Ideally, no sideband component exists around the fundamental component in stator currents. In case a rotor is asymmetrical due to a bar or end ring breakage or a non-oval shape, it causes asymmetrical rotor MMF, and this leads to backward traveling rotor MMF. The backward rotor MMF cause induced voltages in the stator windings with particular frequencies. The comparison between the healthy machine and the asymmetrical rotor machine is presented below;
Healthy Condition
Consider an ideal induction motor. The rotor is perfectly balanced. Rotor bar currents are given by [2];
Figure 3.1 Rotor bar current
2 ( 1)
( ) 1Re( ) cos( ( 1) )r
p njQ
b n b bI I e I s t nπ
ω φ−
−
= = − − (3. 1)
Where, 2
r
pQπφ =
Consequently, the rotor MMF generated by each rotor bar current can be expressed as;
1cos( ( 1) ) cos( ( 1) )bn bF NI s t n nω φ θ φ= − − ⋅ − − (3. 2)
It should be noted that the distribution of rotor bars has been considered in Equation 3.2. Consider only 1 pole pair of the motor in 360 electrical degrees. If there is no breakage, the total rotor MMF at an angle θ can be expressed as;
14
, 1 1
1
1
1
2 2[cos( )cos( ) cos( ) cos( )
2 2 2 2 cos( ) cos( ) ...
( 1) 2 ( 1) 2 cos( )cos( )]
1 cos( )2
b res br r
r r
r r
r r
rb
p pF NI s t s tQ Q
p ps tQ Q
Q Qp pp ps tQ Q
Q NI s tp
π πω θ ω θ
π πω θ
π πω θ
θ ω
= + − −
⋅ ⋅+ − − +
− ⋅ − ⋅+ − −
= ⋅ −
(3. 3)
It can be seen from Equation 3.3 that only the forward MMF is generated by the rotor currents. Assume the airgap length is constant, and slotting effects are neglected. The rotor flux density and induced voltages can be obtained through the following steps;
• The frequency of the rotor MMF; 2 1f sf=
• The electrical rotational frequency; 1(1 )rf s f= −
• The frequency of the rotor airgap flux density;
, 1 1 1(1 )B rf s f sf f= − + =
• The frequency of induced voltages in the stator windings is 1f or the fundamental frequency.
Rotor Asymmetry
Consider an induction motor with one broken rotor bar. The position of the broken bar is placed at the second rotor bar. Due to the defect, currents induced in each rotor bar are not symmetrical. To simplify the analytical expression, each rotor bar current is assumed to be unchanged. (The analytical expression for rotor currents due to broken bars is presented in [32].) Therefore, the total rotor MMF can be determined by subtracting the rotor MMF induced by the second rotor bar current from the ideal rotor MMF;
, , 1
1 1
1 1
2 2cos( ) cos( )
1 2 2 cos( ) cos( ) cos( )21 1 2 2 ( 1) cos( ) cos( )2 2
b res b res br r
rb b
r r
rb b
r
p pF F NI s tQ Q
Q p pNI s t NI s tp Q QQ pNI s t NI s tp Q
π πω θ
π πθ ω ω θ
πθ ω θ ω
′ = − − −
= ⋅ − − − −
⋅= ⋅ − − − + −
(3. 4)
It can be seen from Equation 3.4 that the additional backward MMF exists in the airgap. The frequency of the backward MMF is equal to the forward one. Due to the backward MMF, the induced voltages corresponding to the twice slip frequency is generated around the fundamental component. The frequency of rotor flux density and induced voltages due to the backward MMF can be obtained through the following steps;
Figure 3.2 Speed and direction of rotor MMF for healthy condition (Red; rotor reference frame, Black; stator reference frame)
15
• The airgap flux density due to the backward MMF can induce voltages in the stator windings with the frequency as;
1 1 1 1(1 ) 2brbf s f sf f sf= − − = −
• Due to this component, speed and torque oscillation at the frequency 12sf will be present. From these, the upper sideband component corresponding to the twice slip frequency 12sf above the fundamental component will arise [21,32].
According to [4-9, 15, 19, 21-25], in summary, the indicator of rotor asymmetry is the sideband components around the fundamental. The frequency of the sideband components corresponds to;
1(1 2 )brbf ks f= m , 1,2,3,...k = (3. 5)
In addition, in [4, 5, 10, 19, 20, 32], it has been presented that the consequences of rotor asymmetry can be detected by the components following the equation below;
1[( )(1 ) ]brbkf s s fp
= − ± , 1,5,7,11,13,...kp= (3. 6)
Where, 1f is the fundamental frequency, and s is slip. The analytical explanations in different approaches can be found in [11-13, 31-32].
Unfortunately, if broken bars are located at 180 electrical degrees away from each other, sideband components do not exist [5]. The reason is that the rotor MMF is still symmetrical, so only the forward rotor MMF is generated. Consider an induction motor with two broken bars placed at 180 electrical degrees away from each other. The positions of the broken bars are at the second and the ( 1)rQ p + -th bars. The total rotor MMF can be determined as below.
, , 1
1
1 1
1
2 2[cos( ) cos( )
( 1) 2 ( 1) 2 cos( ) cos( )]
1 cos( ) [cos( )]21 ( 2) cos( )2
b res b res br r
r r
r r
rb b
rb
p pF F NI s tQ Q
Q Qp pp ps tQ Q
Q NI s t NI s tpQ NI s tp
π πω θ
π πω θ
θ ω θ ω
θ ω
′′ = − − −
+ ++ − −
= ⋅ − − −
= ⋅ − −
(3. 7)
It can be seen from Equation 3.7 that only the forward rotor MMF is produced. Therefore, the sideband components do not exist in this case.
In practice, an uneven rotor bar resistance or rotor asymmetry can possibly exist due to manufacture, so sidebands around the fundamental component can be observed even though the machine is healthy. In addition, the amplitude ratio of the first sideband components to the fundamental components is usually chosen as the fault feature to detect a
Figure 3.3 Speed and direction of rotor MMF for rotor asymmetry
16
defected rotor through determining whether the ration exceeds a certain threshold or not. However, there is no standard value for the threshold level.
1f s k 1(1 2 )brbf ks f= m
[Hz] k
p 1[( )(1 ) ]brb
kf s s fp
= − ±
[Hz] 50 0.01 1 49 1 50
51 49 2 48 5 248 52 247 3 47 7 347 53 346
Table 3.1 Detected frequencies on stator current for rotor asymmetry
Furthermore, it is possible to estimate a number of broken rotor bars as stated in [5];
sin2 (2 )
brb
b
II p
απ α
≅−
(3. 8)
And 2
r
npQπα =
Where, brbI = amplitude of the first lower sideband frequency bI = amplitude of the fundamental component n = numbers of broken bars
Another expression for estimation of defected rotor bars presented in [7] is;
20
2
10
rNQn
p=
+ (3. 9)
Where; n = estimate of the number of broken bars N = average dB difference value between the upper and lower sideband and
the fundamental component. p = number of pole pair
3.3 Airgap Asymmetry
Airgap eccentricity is a condition of an unequal airgap that exists between a stator and a rotor. It results in an unbalance magnetic pull (UMP) or unbalance radial forces, which can cause damages in a motor by rubbing between a stator and a rotor. In addition, the radial magnetic force waves can act on a stator core and subject to stator windings unnecessarily and potentially harmful vibration. Therefore, it would be advantageous if airgap eccentricity can be detected before machines are deteriorated. There are three types of eccentricities called static, dynamic and mix of static and dynamic eccentricities. All can be distinguished by the characteristic of the airgap.
The subjects of on-line detection of airgap eccentricities in three-phase induction motors have been proposed by many researchers. In addition to the stator line current, other techniques, such as negative sequence impedance[35], the extend park’s vector [26,27],
17
instantaneous power[5], axial flux[40], inject low frequency signal[39] ,or vibration[38], have been presented for detecting airgap eccentric faults.
3.3.1 Static eccentricity
In the case of static eccentricity, a position of a minimum radial airgap is fixed in space. It causes a steady unbalanced magnetic pull (UMP) in one direction. This can lead to a bent rotor shaft or bearing wear and tear. It can also lead to some degree of dynamic eccentricity.
Static eccentricity can occur when a rotor is displaced from a bore center, but it is still turning upon its bore center [29,30,33]. It can be simplified as shown in Figure 3.4.
3.3.1.1 Cause for static eccentricity
Static eccentricity can be caused by;
1. The oval shape of the stator core due to manufacture
2. Misalignment bearing position due to assembly
3. Bearing wear
4. Misalignment of mechanical couplings
3.3.1.2 Detection of static eccentricity
If a rotor and a stator are assumed to be smooth, the airgap permeance can be expressed of two terms, which are the constant permeance term and the time dependent permeance term due to the rotor rotation as [2,28,29];
0( , ) cos( )t tε ε εθ θ ω ϕΛ = Λ +Λ − − (3. 10)
Where, 0Λ represents a constant airgap permeance, εΛ is a peak of the permeance influenced by rotor eccentricity, εω is an angular frequency of a rotor center relative to a stator and εϕ is a phase angle.
For static eccentricity, the angular frequency εω is zero because the rotor does not rotate around the motor center but spins around its own center. Thus, the airgap permeance function for static eccentricity can be expressed as; 0( , ) cos( )t εθ θΛ = Λ +Λ (3. 11)
By Equation 2.10 and 3.11, the airgap flux density can therefore be determined as shown in Equation 3.12.
Figure 3.4 Static Eccentricity
18
{ }
0 ,1
0 ,1
,1
ˆ( cos( )) cos( )
ˆ cos( )
ˆ cos[( 1) ] cos[( 1) ]2
R
R
R
B F p t
F p t
F p t p t
ε ε μ μμ
μ μμ
εμ μ μ
μ
θ μ θ ω
μ θ ω
μ θ ω μ θ ω
∞
=
∞
=
∞
=
= Λ +Λ ⋅ −
= Λ − +
Λ+ − + − +
∑
∑
∑
(3. 12)
From Equation 3.12, it can be seen that there are two additional terms in the function of the airgap flux density due to the static eccentricity. The frequency of induced voltages influenced from the static eccentricity can be expressed as;
1 2(1 )[ 1]
2ind rsf f g Q
pμωπ
−= = + (3. 13)
Consider influences from time harmonics in supply voltages. From Equation 3.13, the frequency of induced voltages and currents in the stator windings can be expressed as given by [6, 7, 9, 20, and 28];
, 1(1 )[ ]s ecc r
sf kQ n fp−
= ± ⋅ (3. 14)
Where; k = 1, 2, 3… n = order of stator time harmonics present in the power supply feeding the
motor ( 1,3,5...n = )
It can be seen that the expression in 3.14 is actually for rotor slot harmonics. On the other word, the static eccentricity results in a rise of the rotor slot harmonic components. Moreover, the experiments in [28] have shown that the amplitude of the components calculated from Equation 3.14 does not change significantly, when an induction motor is applied by only the static eccentricity. However, the static eccentric variations can result in the introduction of dynamic eccentricity.
3.3.2 Dynamic eccentricity
Dynamic eccentricity occurs when a rotor turns upon a stator bore centre but not its own center. It causes a minimum airgap which is always moving in the airgap. For the case that the rotor center rotates around the motor center with the rotational speed as shown in Figure 3.5, it is called as dynamic eccentricity [4,29,30,33].
In addition, there are other cases of dynamic eccentricity. One of them is that the revolving speed of the rotor center is not equal to the rotational speed [54]. It should be noted that only the type, which is that the rotor axis is parallel to the stator axis, is studied here.
Figure 3.5 Dynamic Eccentricity
19
3.3.2.1 Cause for dynamic eccentricity
As mentioned under the topic of static eccentricity, static eccentricity can lead to some degree of dynamic eccentricity due to UMP. Hence, the causes mentioned in 3.2.1.1 should be valid. In addition, mechanical resonance at critical speed can result in dynamic eccentricity.
3.3.2.2 Detection of dynamic eccentricity
According to Equation 3.10, the revolving speed of the rotor center is equal to the rotational speed for the dynamic eccentricity. That means the angular frequency εω is equal to the mechanical rotational speed mω . Therefore, the airgap permeance function for dynamic eccentricity can be determined as [2,28];
0( , ) cos( )mt tεθ θ ωΛ = Λ +Λ − (3. 15)
Hence, the airgap flux density can be derived from the permeance function in Equation 3.15 and the rotor MMF in Equation 2.10.
{ }
0 ,1
0 ,1
,1
ˆ[ cos( )] cos( )
ˆ cos( )
ˆ cos[( 1) ( ) ] cos[( 1) ( ) ]2
rR
R
r rR
B t F p tp
F p t
F p t p tp p
ε ε μ μμ
μ μμ
εμ μ μ
μ
ωθ μ θ ω
μ θ ω
ω ωμ θ ω μ θ ω
∞
=
∞
=
∞
=
= Λ + Λ − ⋅ −
= Λ − +
Λ+ − + + − + −
∑
∑
∑
(3. 16)
The induced frequency of stator voltages and currents influenced from dynamic eccentricity can be determined from two additional terms in the airgap flux density as shown in Equation 3.17.
1 2
1 (1 )( ) [( 1) 1]2
rind r
sf f g Qp pμωω
π−
= ± = ± + (3. 17)
When time harmonics of supply voltages and dynamic eccentric orders are taken into account, Equation 3.17 can be modified as given by [6, 7, 9, 20, and 28];
, 1(1 )[( ) ]d ecc r d
sf kQ n n fp−
= ± ± ⋅ (3. 18)
Where; k = 1, 2, 3… dn = dynamic eccentric order ( dn =1, 2, 3,) n = Time harmonic order of supply voltages driving motors ( 1,3,5...n = )
According to [4,28], in the case when one of these harmonics influenced from static or dynamic eccentricities is a multiple of three, it may not exist theoretically in the line currents of a balance three phase machine. Besides, induction motors corresponding to the relationship in Equation 3.19 are ascertained to generate principle slot harmonics, but they will not give rise to these harmonics with only static or dynamic eccentricities. 2 [3( ) ]rQ p m q r= ± ± (3. 19)
20
However, only a particular combination of machine pole pairs and numbers of rotor slots will give a significant rise of only static or dynamic eccentricities related to components. The relationship for a three-phase integral slot and 60-degree phase belt machine is given by [4,28];
2 [3( ) ]rQ p m q r k= ± ± ± (3. 20)
Where, 0,1,2,3,...m q± = 0 1r or= 1 2k or=
3.3.3 Mix eccentricity
In reality, both static and dynamic eccentricities tend to co-exits in machines. With this condition, a rotor turns around neither its bore center nor a stator bore center, but it revolves around a point between the stator and rotor centers. This condition can be presented by Figure 3.6 showing that the rotational center or the motor center can be anywhere between the stator and rotor centers [29].
3.3.3.1 Detection of Mix eccentricity
According to Equation 3.10, 3.11 and 3.15, the permeance function of mix eccentricity can be determined as three different terms that are influences from the constant permeance, static eccentric and dynamic eccentric terms [29];
0 1 2( , ) cos( ) cos( )mt tθ θ θ ωΛ = Λ +Λ +Λ − (3. 21)
It can be seen from Equation 3.21 that the airgap permeance for mix eccentricity results from both static and dynamic eccentricities. In addition to the influences corresponding to Equation 3.14 and 3.18, the consequence from both can be found as the results of amplitude modulation. The low frequency components corresponding to the rotational frequency will exist around the fundamental frequency. The expression for the low frequency components is shown in Equation 3.22. The analytical study to derive this expression can be seen in [29].
, 1 11[1 ( )]mix ecc r
sf f mf f kp−
= ± = ± (3. 22)
Where; , 1, 2,3,...m k p =
It should be noted that the formula in Equation 3.22 is for the case that the revolving speed of the rotor center is equal to the rotational speed. In addition, it can be observed that the low frequency components are placed away with a multiple of the rotational frequency from the fundamental component.
Figure 3.6 Mix eccentricity
21
Static Ecc. Dynamic Ecc. Mix Ecc. 1f p s rQ
k n,s eccf
[Hz] k dn n,d eccf
[Hz] m ,mix eccf
[Hz] 50 3 0.01 68 1 1 1172 1 1 1 1188.5 1 66.5
1072 1088.5 33.5 5 1372 1155.5 2 83 872 1055.5 17 1 2 1 1205 1105 1139 1039
Table 3.2 Detected frequencies on stator current for airgap asymmetry
3.4 Stator Asymmetry
According to the surveys [58], the majority of failure related to a motor stator is breakdown of the turn-to-turn insulation. Although the induction motor can still run when some of the turns are shorted, they can consequently lead to damages on adjacent coils and a stator core, so that a ground fault can occur. To reduce repairing costs and outage time due to the stator winding fault, the early detection of inter-turn short circuit is useful.
3.4.1 Cause for Inter-Turn Short Circuit
There are many reasons that can cause the degradation on the stator insulation. The causes can be summarized as [4, 36];
1. Thermal stresses due to thermal ageing and thermal overloading: For the thermal ageing, it is a result from the operating temperature. As known, the insulation life gets half for every 10o k increase in temperature. To cope with the thermal ageing due to the temperature in the windings, reducing the operating temperature or increasing the class of insulation materials can be applied. Thermal overloading can be caused by the applied voltage variations, unbalanced phase voltage, cycling overloading, obstructed ventilation, higher ambient temperature, etc. All of these can increase the temperature and can initiate the thermal stress in the machine.
2. Electrical stresses due to voltage stresses in the windings: The voltage stress in the windings can be caused by having a void in the insulation, which can cause the partial discharge. In addition, the surge on electrical supply system can initiate the voltage stresses in the windings as well.
3. Mechanical Stresses: These stresses might be due to coil movement, which is a result from the force inside the machine, and rotor striking the stator, which is caused from many reasons, such as bearing failures, shaft deflection, rotor-to-stator misalignment, etc.
4. Environmental stresses/Contamination: the winding insulation can be deteriorated by chemicals, such as oil, moisture or dirt, etc.
5. Ageing: the winding insulation can be degraded by time.
22
3.4.2 Detection of Inter-Turn Stator Winding Fault
The early technique to detect the stator fault is the partial discharge technique [36]. Axial leakage flux monitoring [42], negative sequence impedance or negative sequence current [35,46] and zero-sequence component [37,41] have been presented as method of detection for the inter-turn short circuit fault in an early state.
The effect of the inter-turn short circuit fault is that some turns from stator windings are removed. This causes a small but finite effect on the airgap flux density. When a short circuit happens, phase windings have less numbers of turns, so they produce less MMF. Moreover, the currents that flow in the shorted windings also produce MMF, which is opposite to and against the main MMF produced by the phase windings [42,43,44,54].
Figure 3.7 Diagram of inter-turn short circuit in one section of a single phase coil winding
For MCSA, the induced frequency resulting from inter-turn stator fault has been presented in [6] as expressed in Equation 3.23.
1 (1 )st
nf f s kp
= ⋅ − ± (3. 23)
Where; n = 1, 2, 3… k = 1, 3, 5…
In addition, the other two expressions considering influences from saturation in materials as well as influences from different sources have been presented in [43]. The first expression originating from the stator currents is as;
1(1 ) 2st r
sf f kQ m np
⎧ ⎫−= ± ±⎨ ⎬
⎩ ⎭ (3. 24)
The second expression originating from the rotor currents is as;
1
(1 )( ) 2st rsf f kQ i m n s
p⎧ ⎫−
= ± ± ± ⋅⎨ ⎬⎩ ⎭
(3. 25)
Where; i = 1,2,3,… n = 1,2,3,… k = influences from the rotor slots = 1,2,3,… m = influences from the saturation = 0,1,2,3,…
23
1f p s rQ n k stf
[Eq(3.23)] [Hz]
k m nstf
[Eq (3.24)] [Hz]
k i m n stf
[Eq (3.25)] [Hz]
50 3 0.01 68 1 1 66.5 1 0 -1 1072 0 1 0 1 17 1 1 33.5 1 0 1 1172 0 2 0 1 33.5 2 1 83 1 1 1 1272 0 3 0 1 50 1 3 17 1 2 1 1372 0 4 0 1 66.5 1 5 266.5 0 5 0 1 83 1 5 233.5
Table 3.3 Detected frequencies on stator current for inter-turn short circuit
It can be observed that the expression in 3.24 is similar to the expression for rotor slot harmonics. In addition, the expressions in 3.23 are similar to Equation 3.25 if influences from the saturation in material are not taken into account. These imply that the inter-turn short circuit fault makes influences on line current by rises of rotor slot harmonics and the components corresponding to the rotational frequency. However, these are not sufficient to identify the mix eccentricity and the inter-turn short circuit fault.
To separate both faults, current amplitude of each phase and phase shifts between each phase current are required. In the healthy condition, impedances of each phase winding are normally balanced, so the current amplitude of each phase is also balanced. The phase shift between each phase is 120o . Due to defected turns, impedances of the three phase windings become unbalanced, so the three phase currents will be unbalanced as well. The phase shift between each phase will be also distorted from 120o . Moreover, the third harmonic contents will become dominant. Therefore, information on amplitude, phase shift and the third harmonic contents is necessary for separating the inter-turn short circuit fault from the mix eccentricity.
3.5 Bearing Damages
Bearing is the part used to hold a rotor shaft of induction motors. Faults on bearing may result in increasing vibration and noise levels. Bearing faults can also cause some damages on mechanical couplings that connect to a rotor shaft. To protect motors and mechanical couplings, detection of bearing faults in an early state becomes useful.
3.5.1 Cause for Bearing Damages
The bearing damages can result from many causes; internal causes, such as induced bearing currents due to an unbalanced rotor or external causes, such as grease. However, they can be summarized as;
1. High vibration due to foundations, mechanical couplings or loads
2. Inherent eccentricities, which cause unbalance magnetic force
3. Bearing current which cause an electrical discharge or sparking in bearings
4. Contamination and corrosion which is caused by pitting and sanding action of hard and abrasive minute particles or corrosive action of water, acid, dirt etc
5. Improper lubrication including both over and under lubrication causing heating and abrasion
24
6. Improper installation of bearing; by improperly forcing bearings onto a shaft or in a housing (due to misalignment) indentations are formed in the raceways.
3.5.2 Detection of Bearing Damages
Bearing faults can be detected by the increased vibration in the high frequency spectrums [36]. However, the cost of obtaining the vibration measurement is routinely high due to the measurement equipment. Instead, the stator-current-based monitoring scheme is inexpensive because it requires no additional sensors.
In [4,5,47,48], it has been suggested that bearing faults can be caused by mechanical displacements in the airgap. They can manifest themselves as a combination of rotating eccentricity moving in both directions. This can result in the increased bearing vibrations, and the bearing vibrations can reflect themselves in the currents regarding the components as;
1mech vf f mf= ± (3. 26)
Where, m = 1,2,3,… vf = the characteristic vibration frequencies based upon the bearing
dimensions.
Generally, the majority of electrical machines use ball or rolling element bearings. These bearing types consist of two rings; inner and outer rings. Damages on these bearing types can be categorized into four different damages [4,5,48];
1. Damage on an outer bearing race: the vibration frequency is as;
( ) [1 cos( )]2v rN BDf f
PDβ= − (3. 27)
2. Damage on an inner bearing race : the vibration frequency is as;
( ) [1 cos( )]2v rN BDf f
PDβ= − (3. 28)
3. Damage on a ball: the vibration frequency is as;
21 [ cos( )]rv
PD f BDfBD PD
β⋅ ⎧ ⎫= −⎨ ⎬⎩ ⎭
(3. 29)
4. Damage on a train: the vibration frequency is as;
2( ) [1 cos( )]
2r
vf BDf
PDβ= − (3. 30)
Where; N = number of bearing balls BD = ball diameter PD = ball pitch diameter β = The contact angle of the ball with the races
Figure 3.8 Ball Bearing Dimension
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3.6 Mechanical Coupling
It has been presented in [7] that mechanical equipments, such as gearbox etc. can influence on stator currents as sideband components around the fundamental frequency. Their frequency corresponds to their rotational speed. Therefore, it is necessary to take the influences from the mechanical couplings into consideration when an investigation on stator currents is preformed. Information of mechanical systems is required in order to make a proper investigation.
1 ,mech r mf f mf= ± (3. 31)
Where, ,r mf is the rotational frequency of the mechanical coupling equipment.
3.7 Oscillating Loads
Influences from oscillating loads on stator currents have been presented in [49]. The experiment was performed by running an induction motor with the 10Hz periodical load torque with the 50% duty cycle. The sidebands placed at 10Hz away from the fundamental component, were found. It should be noted that influences from oscillating loads may lead to the wrong conclusion. Therefore, this point should be considered when one makes a diagnosis.
With the assumptions that induction motors are lossless and are fed by the perfect sinusoidal supply voltages, it is possible to consider the effects of the oscillating loads in stator currents. The input currents can be made up of the sum of the components from the fundamental frequency and the influences from the oscillating loads, which can reflect themselves as the sidebands as shown in Equation 3.32.
1 1 2
1 1 2
1 1 2
cos(2 ) cos(2 )2 2cos(2 ) cos(2 )3 3
2 2cos(2 ) cos(2 )3 3
a sb
b sb
c sb
I I f t I f t
I I f t I f t
I I f t I f t
π φ π φπ ππ φ π φ
π ππ φ π φ
= − + −
= − − + − −
= + − + + −
(3. 32)
Hence, the input power can be obtained from the product of the input currents and voltages.
[ ]1 1
3 3 cos cos 2 ( )2 2
a a b b c c
sb sb
P V I V I V I
VI VI f f tφ π φ
= ⋅ + ⋅ + ⋅
= ⋅ + ⋅ − − (3. 33)
Where; 1I = Amplitude of the current from the fundamental frequency 2I = Amplitude of the current from the sideband components 1f = The fundamental frequency sbf = The sideband frequency V = Amplitude of the supply voltage
From Equation 3.33, it can be seen that the input power is not constant but pulsates at the frequency 1 sbf f− . On the other word, if the loads pulsate with the frequency 1 sbf f− , the frequency component at sbf can present in the stator currents.
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3.8 Conclusion
The consequences and detection on stator currents for each fault type have been summarized in the Table below;
Fault Consequences Detection
Broken Rotor Bars & End Rings
• Backward rotor MMF due to unsymmetrical rotor currents
Equation 3.5 and 3.6
Static eccentricity • Steady unbalanced magnetic pull due to a space-fixed minimum radial airgap
Rotor Slot Harmonics Equation 3.14
Dynamic eccentricity
• Unbalanced magnetic pull due to periodical minimum radial airgap length
Equation 3.18
Mix eccentricity • Unbalanced magnetic pull influenced from static and dynamic eccentricities
Equation 3.14,3.18 and 3.22
Inter-turn Short Circuit • Unsymmetrical stator MMF and windings
Equation 3.14, 3.23
Bearing Damages • Increased vibrations
Equation 3.26-3.30
• Influences from Mechanical couplings can also reflect themselves in stator currents as the components, of which frequency corresponds to their rotational speed.
• Oscillating loads can cause sideband components in stator currents, of which frequency corresponds to the load frequency.
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4 INSTANTANEOUS POWER APPROACH & THE EXTENDED PARK’S VECTOR APPROACH
4.1 Introduction
In this chapter, two alternative approaches are introduced. The first approach is Instantaneous Power Approach (IPA). The advantage of IPA is that the harmonics can be more easily separated from the fundamental component. The sideband components, which are placed around the fundamental component in the stator currents, will instead appear around DC and twice-fundamental frequency in the instantaneous power. With MCSA, it is difficult to filter out the fundamental component without affecting any sideband components in the stator currents. In contrast, separating the sidebands from DC is much easier by DC compensation. In addition, the instantaneous power also contains more information and has stronger tolerance on distortions than the stator currents since the instantaneous power is a product of multiplying of the voltage and current. [5,9,49,50]
The other technique is the Extended Park’s Vector (EPVA). This technique is to consider the three phase currents in the terms of the d-axis and q-axis components. By this, two indicators, which are Lissajou’s curve and current modulus, will be obtained. By monitoring deviations of an acquired Lissajou’s curve from an expected one, faulty conditions can be easier detected without any profound knowledge requirement. However, the Lissajou’s curve cannot clearly identify what a fault type is. The current modulus is required for fault identification. Similar to IPA, the sideband components are also converted to appear around DC in the current modulus. [14,24-27,45]
4.2 Instantaneous Power Approach
Healthy Condition
Begin with one phase instantaneous power of an ideal induction motor. The expression can be derived from ideal supply voltages and currents as shown in Equation 4.1.
( ) ( ) ( )p t v t i t= ⋅ (4. 1)
Thus, instantaneous power can be expressed as;
1( ) [cos(2 ) cos( )]rms rmsp t V I tω ϕ ϕ= − + (4. 2)
Where, ( )v t = phase voltages (L-N or L-L) = 12 cos( )rmsV tω
( )i t = line currents = 12 cos( )rmsI tω ϕ− ϕ = a load angle
It can be seen from Equation 4.2 that the instantaneous power consists of 2 terms; DC and the sinusoidal term with the twice-fundamental frequency. The former represents the real power, and the latter represents the apparent power. Moreover, some additional components on the instantaneous power caused by interactions of the first three harmonics of supply voltages and currents are also present at the frequencies 1 1 1 1 1, 3 , 4 , 5 , and 6f f f f f [49].
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Faulty condition
According to the previous chapter, the stator currents contain some additional components due to an abnormality in induction motors. For simplicity, it can be assumed that the additional components generated by faults result from the amplitude. Thus, the stator currents in the abnormal condition is expressed as shown in Equation 4.3 [9,49,50].
{ }1 1
( ) ( ) [1 cos( )]
( ) cos[( ) ( )] cos[( ) ( )]2
M f
rmsf f
i t i t M t
I Mi t t t
ω β
ω ω ϕ β ω ω ϕ β
= ⋅ + −
= + + − + + − − − (4. 3)
Where, M is a modulation index, fω is a modulation angular frequency caused by the abnormality, and β is a modulation phase. Hence, the modulated instantaneous power can be expressed as;
1
1
( ) ( ) {cos[(2 ) ( )]2
cos[(2 ) ( )] 2cos( ) cos( )}
rms rmsM f
f f
MV Ip t p t tω ω ϕ β
ω ω ϕ β ϕ ω β
= + + − +
+ − − − + − (4. 4)
From Equation 4.4, it can be seen that the sideband components are still present in the instantaneous power, but they are converted to place at fω around DC and 12 fω ω± around
the twice fundamental frequency. The components placed around DC, subsequently called characteristic components, provide an extra piece of diagnostic information about the health of the motor.
According to the previous chapter, the fault indicators for IPA can be derived from the particular components influenced from faults on stator currents as shown in Table 4.1.
Condition mf IPA Expected components
Rotor asymmetry 12ksf 12ksf , 1 12 2f ksf±
Mix eccentricity rmf rmf , 12 rf mf±
Inter-turn short circuit [Eq (3.23) when k=1] 1(1 ) r
n s f nfp
− = rnf , 12 rf nf±
Table 4.1 Detected frequencies in the instantaneous power for different types of faults
From Table 4.1, it can be seen that the IPA cannot separate the mix eccentricity and the inter-turn short circuit fault. However, the consequences of the latter, which results in increasing the third harmonic contents in stator currents, have to be considered. These can cause the components at 1 1 1 12 ,4 ,8 and 10f f f f to exist on the instantaneous power spectrums.
However, IPA does not gain any advantage to detect static and dynamic eccentricities. In addition, since IPA is based on information from the stator currents, noise on the stator currents also still exist on the instantaneous power. This can cause some difficulty to detect the particular components when fault severity is small. Nevertheless, IPA is still better than MCSA for detecting small particular components since the power amplitude is normally much higher than the current amplitude. By filtering out DC, the small particular components can show themselves to be significant.
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4.3 The Extended Park’s Vector Approach
Park transformation is used to transform stator currents from the three-phase system (A-B-C) to the two-phase system (D-Q). The expression for transformation is as presented by [5,14,24,25,27,45];
2 1 13 6 6
1 12 2
d a b c
q b c
i i i i
i i i
= − −
= −
(4. 5)
In addition, the expression for the current modulus is as;
d qCurrent Modulus i ji= + (4. 6)
It should be noted that the transformation is based on the stator reference frame.
Healthy condition
Under a healthy condition, the three phase currents can be expressed as shown in Equation 2.2. Therefore the d-axis and q-axis currents can be determined as;
1
1
6 sin( )26 sin( )
2 2
d
q
i I t
i I t
ω
πω
=
= −
(4. 7)
The Lissajou’s curve represents the function between the d-axis and q-axis components as ( )q di f i= . From Equation 4.7, the Lissajou’s curve for the healthy induction motor has a
perfect circular shape with the center at the origin, and its diameter is equal to ( 6 / 2)I as can be seen in Figure 4.1(a). Since the diameters of the Lissajou’s curve are proportional to the current amplitude, the shape becomes thicker when motor loads are changing. In addition, from Equation 4.6 and 4.7, the current modulus for the healthy condition contains only DC.
Faulty condition
In a faulty condition, due to the particular components influenced from faults on stator currents, the shape of Lissajou’s curve becomes distorted. In [14,24], detection of rotor asymmetry by monitoring the Lissajou’s curve has been presented. The rim of the Lissajou’s curve becomes thicker when the rotor is asymmetrical. For example, the Lissajou’s curve for 10-broken rotor bars shown in Figure 4.1(b). This is one of advantages, which allows the detection of faulty conditions by monitoring the deviations of the acquired patterns. In addition, the analytical expression for the rotor asymmetry has been derived by [25], and the results have shown that the sideband components in the stator currents influenced from the rotor asymmetry could be transformed to place at the frequency 1 12 , 4sf sf around DC in the current modulus.
In [26, 27], it has presented that the Lissajou’s curve in the case of eccentricity is quite similar to the healthy one. However, it becomes a bit thicker, when a high degree of
30
eccentricities takes place. This can imply that the Lissajou’s curve cannot detect the eccentricities.
According to the previous chapter, the signature of mix eccentricity is the sidebands corresponding to the rotational frequency in the stator currents. Similar to rotor asymmetry, these components are also transformed to be placed at the frequency , 2r rf f around DC in the current modulus. However, EPVA does not give any advantage to detect static and dynamic eccentricities.
Figure 4.1 Lissajou’s curve for various conditions (FEM Simulation)
To detect the inter-turn short circuit fault, both the Lissajou’s curve and the current modulus have to be determined. In the healthy condition, the stator current contains only the positive sequence, so the circle shape of the Lissajou’s curve is still valid. However, under an abnormal condition, since phase impedances are unbalance due to the defected windings, they cause unbalanced supply currents and induces the negative sequence. Due to the negative sequence, the Lissajou’s curve can show some distortion as an elliptical shape. For example, Lissajou’s curve for the 6-inter-turn short circuit fault shown in Figure 4.1(d). In addition, due to the existence of the negative-sequence, it manifests itself in the current modulus by the component at the twice-fundamental frequency as shown in Figure 4.2 [45,52].
In table 4.3, the summarized table for fault indicators by EPVA is presented. It should be noted that the Lissajou’s curve is not effective to recognize an abnormal condition, of which the fault severity is small. Therefore, it is necessary to determine both the Lissajou’s curve and the current modulus.
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a) The Lissajou’s curve [45] b) The Park’s vector modulus [45]
Figure 4.2 Relationship between the symmetrical components and Park’s vector for stator asymmetry
Condition The Lissajou’s curve Spectrum of Park’s modulus Healthy Circle DC Broken Rotor Bars or End Rings Circle, Thicker DC, 12sf , 14sf
Mix eccentricity Circle (Thicker for high degree of eccentricities) DC, rf , 2 rf
Stator Winding Fault Ellipse DC, rf , 2 rf , 12 f
Table 4.2 Detected Lissajou’s curves and spectrum of park’s modulus for each type of faults
4.4 Conclusion
The Instantaneous Power Approach and The Extended Park’s Vector Approach have been presented in this chapter. The advantage of IPA is to convert the characteristic components to appear around DC and twice-fundamental frequency. By easily filtering out DC, the characteristic components can show themselves more clearly. However, the disadvantage of this approach is that it is not sufficient for static and dynamic eccentricities. It also requires additionally one phase voltage. The summary of fault indicators by IPA can be seen in Table 4.1.
There are two indicators for EPVA; Lissajou’s curve and current modulus. Faulty conditions can be easily detected by monitoring deviation in the Lissajou’s curve. However, in order to identify the fault type, both indicators have to be determined. The drawback of this approach is that it is not effective for static and dynamic eccentricities either. It also requires two phase currents additionally. The summary of fault indicators by EPVA can be seen in Table 4.2.
In addition, both approaches require information on three phase rms currents so that they are able to separate the inter-turn short circuit fault from mix eccentricity.
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5 FEM SIMULATION MODELS
5.1 Introduction
In this chapter, it is presented how the fault can be implemented in FEM models with FLUX2D and what the simulation results are. Models of six fault types categorized in to three fault groups have been built up. First, rotor asymmetry caused by broken rotor bars or broken rotor end rings is described. Second, airgap asymmetry resulting from static, dynamic, and mix of static and dynamic eccentricities is explained. Then, stator asymmetry caused by an inter-turn short circuit fault is introduced.
The chapter begins with details about FEM models in general. Next, the physical modeling of an induction motor is introduced. Then, the implementations of the six fault types are discussed. In all cases, the characteristics of each fault type presented in Chapter 3 are applied to form and verify FEM models.
5.2 Finite element modeling
A FEM model is formed by three main parts. The first part is geometry of a studied induction motor. The second is an electrical circuit, which represents connections, couplings and electrical parameters. The last part includes material properties, such as electric and magnetic characteristics.
Usually, it is sufficient for a healthy induction motor to consider only 1 pole due to its symmetry regarding the electric and magnetic phenomena within each pole. However, it is not valid for a faulty machine due to asymmetry with regard to the electric and magnetic phenomena. Hence, it requires studying a fully detailed physical model of an induction motor. The studied induction motor is HXR400LD6. Its specifications can be found in Table 5.1.
5.3 Physical Model of Induction Motors
As mentioned above, three main parts for forming up a physical model are geometry, circuit and material properties. However, the material properties for the studied motor are not presented here. The geometry for the studied induction motor is shown in Figure 5.2.
Figure 5.1 Steps to model the problem
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(a) (b)
Figure 5.2 Geometry of the studied induction machine in the healthy condition
The geometry represents the real dimension of the motor and also contains information on its windings; i.e. pitch factor. The most important thing regarding the geometry is the airgap.
Due to the Time-Stepping simulation, the so-called, “Rotating airgap” is required. The rotating airgap can be either equal to or less than the actual airgap. In order to get better accuracy in the simulation results, the real airgap should contain several layers, but only one layer of the airgap is required to be the rotating airgap. Figure 5.3 shows two types of the rotating airgap; one airgap layer and several airgap layers. Moreover, the rotating airgap is required to be uniform in length around the origin point (0, 0). [53]
The circuit shown in Figure 5.4(a) is composed of two main parts. The first part represents the stator circuit as shown in Figure 5.4(b). The stator circuit is formed by three phase voltage sources, coils and end winding resistances and inductances. The coils represent the winding resistance and inductance as well as the coupling of the electric and magnetic phenomena. In addition, the stator circuit also contains information about the winding connection. Here is “Star-connection” for the studied induction motor.
The second part describes the rotor circuit as shown in Figure 5.4(c). It is composed of 2 main parts. The first part is all vertical components representing rotor bars. Each rotor bar is modeled by a resistance, an inductance and a coil. The resistance and inductance describe the electrical characteristics. The coil is for the electric and magnetic coupling. The other part is all horizontal components representing rotor end rings. The coil is not required for the model of rotor end rings because there is no coupling.
Figure 5.3 Two definitions for the rotating airgap
34
Moreover, the magnetic coupling between the stator and rotor circuits is represented by the line linking between both circuits. The details of the circuit parameters are described in Table 5.2 and 5.3.
Specification Induction Motor
Model HXR400LD6 Number of poles 6 Number of phase 3
Number of parallel paths 1 Number of stator slot 54 Number of rotor slots 68
Connection Star Rated voltage [V] 6000
Rated frequency [Hz] 50 Rated current [A] 43.467 Rated power [kW] 350
Number of conductor in a half slot slotN Pitch factor 8/9
Table 5.1 Specifications of the studied induction motor
Component Definition Parameter
Ua Supply Voltage, A Phase (rms) Ub Supply Voltage, B Phase (rms) Uc Supply Voltage, C Phase (rms)
3464.1 V
SS1AP1P1 Coil A (+), representing the winding turns in the A-phase slot, Current direction (+)
SS1AM1P1 Coil A (-), representing the winding turns in the A-phase slot, Current direction (-)
SS1BP1P1 Coil B (+), representing the winding turns in the B-phase slot, Current direction (+)
SS1BM1P1 Coil B (-), representing the winding turns in the B-phase slot, Current direction (-)
SS1CP1P1 Coil C (+), representing the winding turns in the C-phase slot, Current direction (+)
SS1CM1P1 Coil C (-), representing the winding turns in the C-phase slot, Current direction (-)
N = ,total windingsN turns
R = ,total windingsR Ohm
Stacking Factor = . .S F
STAR1_1 End Winding Resistance, A Phase STBR1_1 End Winding Resistance, B Phase STCR1_1 End Winding Resistance, C Phase
R = ,total endR Ohm
STAL1_1 End Winding Inductance, A Phase STBL1_1 End Winding Inductance, B Phase STCL1_1 End Winding Inductance, C Phase
L = ,total endL H
Table 5.2 Descriptions of the components in the stator circuit
35
Component Definition Parameter
RBAREND(X) Rotor bar resistance of the rotor bar No. X R = barR Ohm.
LBAREND(X) Rotor bar inductance of the rotor bar No. X L = barL H.
RBAR(X) Solid Coil, the rotor bar No. X, (1 turns) -
RRING(X)_1 Rotor end ring resistance of the rotor end ring No. X R = ringR Ohm.
LRING(X)_2 Rotor end ring inductance of the rotor end ring No. X L = ringL H.
(X) Index of rotor bar and rotor end ring 1 to rQ
Table 5.3 Description of the components in the rotor circuit
Figure 5.4 Circuit model of the induction motor
(a) Complete circuit, (b) Stator circuit, (c) Rotor circuit
(a)
(b)
(c)
36
5.4 Rotor Asymmetry
5.4.1 Broken Rotor Bar
Figure 5.5 Circuit model of broken rotor bars
According to Chapter 3, broken rotor bars can cause the unsymmetrical rotor current distribution. This results in the distortion in the magnetic field [10,13,17-19]. Ideally, rotor bar resistance is low, so the currents resulting from induced voltages by the stator flux density can flow through all rotor bars. Under the abnormal condition, cracks or breakages can cause increased rotor bar resistance, so very little or no current can flow in defected bars. The FEM model for rotor broken bars can be developed by 2 different approaches defined as Model 1 and Model 2.
The first model is to eliminate the element, which represent the broken rotor bar in the rotor circuit. This also requires modifying the material properties for the broken bar. Figure 5.6 and Table 5.4 show the modified rotor circuit and material properties for the defected bar.
The second model is to change only the rotor bar resistance to a high value without modifying the rotor circuit and material properties; i.e. 1 6 R e= Ω . This will force a low current flow in the broken rotor bar.
The rotor current density and the normal component of the airgap flux density with one broken rotor bar are shown in Figure 5.7 and 5.8 respectively. It can be seen that the induced current in the broken rotor bar is very low, and this cause the unsymmetrical rotor current distribution. Consequently, the airgap flux density is distorted from the healthy one. In addition, to compare the degree of severity, the model with two-broken rotor bars has been investigated. The normal components of the airgap flux density for the cases with one and two broken rotor bars are compared in Figure 5.9.
Normal Rotor Bar Broken Rotor Bar Iso Mu 1 1
Iso Rhm 2.72E-08 1.00E+6
Table 5.4 Material properties for rotor bars
Figure 5.6 Modified circuit for 1 broken rotor bar
37
Figure 5.7 Rotor current density for 1-broken rotor bar
Figure 5.8 Normal components of airgap flux density for the healthy and 1- broken rotor bar condition
38
Figure 5.9 Normal components of airgap flux density for various fault severity
5.4.2 Broken Rotor End Ring
Figure 5.10 Circuit model of broken end ring
Ideally, there is no circulating current flowing in the rotor due to the complete end rings, all of which are perfectly connected to make a short circuit in the rotor. However, when a part of rotor end rings is broken, this causes non-zero circulating currents flowing in the rotor as shown in Figure 5.10. To obtain the circulating currents, it is possible to modify the rotor circuit by breaking the complete short circuit of the rotor end rings.
The approaches to create the FEM simulation model are similar to those of broken rotor bars. As shown in Figure 5.11, the first approach is to remove the end ring resistance and inductance, which represents the defected rotor end ring. However, its material properties do not require changing. The second one is to increase the end ring resistance; i.e. 20 R = Ω
39
The rotor current density and the normal component of the airgap flux density with one broken rotor end ring are shown in Figure 5.12 and 5.13 respectively. It can be seen that the rotor current density for one broken end ring is distorted from the healthy one due to the non-zero circulating currents. This causes the distortion in the airgap flux density.
Figure 5.12 Rotor current density for healthy and 1-broken end ring conditions
Figure 5.11 Modified circuit for 1-broken end ring
40
Figure 5.13 Normal components of airgap flux density for healthy and one-broken end ring conditions
5.5 Airgap Asymmetry
5.5.1 Static Eccentricity
The main characteristic of static eccentricity is the presence of a space-fixed minimum radial airgap. This can cause a steady unbalanced magnetic pull (UMP) in one direction. The model of static eccentricity can be developed by modifying only the geometry.
The geometry has three different centers; motor, stator and rotor centers. Ideally, all the centers are placed at the same position, so the airgap is symmetrical. By shifting the stator center or the stator geometry away from the motor and rotor centers, the space-fixed minimum airgap will be formed as shown in Figure 5.14.
Where, δ = symmetrical airgap length
ε = eccentric level
Figure 5.14 Geometry for static eccentricity
41
Figure 5.15 Geometry of an induction machine for static eccentricity
The normal component of the airgap flux density for the healthy and 40% static eccentric conditions are compared in Figure 5.16. For the healthy condition shown in Figure 5.16(a), the peak amplitudes of the airgap flux density are in the same level, but this is not valid in the case of static eccentricity either. However, the position of the first highest peak as well as the second peaks is fixed as can be seen from Figure 5.16(b) and (c) even time is varied. By this, it is possible to conclude that the minimum radial airgap is fixed in space and time.
42
Figure 5.16 Normal components of airgap flux density
(a) Healthy condition (b) Static Ecc. at t1 (c) Static Ecc. at t2
43
5.5.2 Dynamic Eccentricity
The main characteristic of dynamic eccentricity is that a position of minimum airgap length always revolves around a motor center. The studied dynamic eccentric case is that the revolving speed of the rotor center around the motor center is equal to the rotational speed. This will cause the periodically changing airgap. The model for dynamic eccentricity can be achieved by shifting the rotor center away from the motor and stator centers as shown in Figure 5.17.
Since the rotor center revolves around the motor center with the rotational speed, the minimum rotating airgap period can be calculated as;
1
2(1 )r
p pTs f
πω⋅
= =−
(5. 1)
Where; p = number of pole pair rω = electrical rotor angular speed s = slip 1f = frequency of supply voltage
In Figure 5.18, the airgap at the particular position for varied time is presented. The simulation of 40% dynamic eccentricity is performed at 1% slip, so the period time is about 0.06 seconds. It can be observed that at the specific position, the minimum airgap takes place every 0.06 seconds. Moreover, at every half of the period time, the maximum airgap will happen instead.
(a) t = 0.06 sec. (b) t = 0.08 sec. (c) t = 0.10 sec. (d) t = 0.12 sec.
Figure 5.18 Airgap at varied time for dynamic eccentricity
Figure 5.17 Geometry for dynamic eccentricity
44
The normal component of the airgap flux density for varied time is compared in Figure 5.20. It can be seen in Figure 5.20(a) and (b) that the airgap flux density at t = 0.06 and 0.12 seconds look similar. The positions of the first and second highest peaks are placed around the same position. The airgap flux density at t = 0.09 and 0.15 seconds are also similar. However, the highest peak changes be the lowest at every half of the period time as can be seen by comparing Figure 5.20(a) and (c) or 5.20(b) and (d). Thus, it is possible to conclude that the minimum radial airgap revolves around the motor center and the airgap length changes periodically.
5.5.3 Mix of Static and Dynamic Eccentricities
The main characteristic of mix eccentricity is that the rotor turn around neither its bore center nor the stator bore center, but it revolves around a point between the stator and rotor bore centers. This results in an unsymmetrical airgap which changes periodically. The difference between dynamic eccentricity and mix eccentricity is that the airgap length for the dynamic eccentricity is changing from the maximum to minimum at every half of the period time, but this is not valid for the mix eccentricity. This can be seen clearly in Appendix I.
Thus, the model for mix eccentricity can be obtained by shifting both stator and rotor centers in order to have the motor center placing between both as shown in Figure 5.19.
Where, δ = symmetrical airgap length
1ε = static eccentric level 2ε = static eccentric level
In Table 5.5, the calculated airgap length for varied time for the 40% dynamic eccentricity and the mix of 15% static and 25% dynamic eccentricities are presented. The Matlab code for the calculation can be found in Appendix II.
Airgap length for dynamic ecc. [mm] Airgap length for mix of ecc. [mm] Time [s] Zeta =0 Zeta = 180 Zeta =0 Zeta = 180 t = 0 1.0800 2.5200 1.0800 2.5200 t = 0.06 1.0814 2.5186 1.0809 2.5191 t = 0.12 1.0857 2.5143 1.0836 2.5165 t= 0.09 2.5168 1.0832 1.9780 1.6220 t= 0.15 2.5112 1.0889 1.9745 1.6256
Table 5.5 Calculated airgap length for dynamic and mix eccentricities
The normal component of the airgap flux density for the mix eccentricity at varied time is shown in Figure 5.21. It can be seen from Figure 5.21(a) and (b) that the positions of the highest peaks in both cases are placed around 0oZeta ≅ . In Figure 5.21 (c) and (d), the positions of the highest peaks in both cases are located around 180oZeta ≅ . In addition, the
Figure 5.19 Geometry for mix eccentricity
45
different amplitude between the peaks placed around 0oZeta ≅ and 180oZeta ≅ in Figure 5.21(a) and (b) is bigger than that in Figure 5.21 (c) and (d). This agrees with the calculated airgap length in Table 5.4. Therefore, the FEM model of the mix eccentricity is verified.
46
Figure 5.20 Normal components of airgap flux density for dynamic eccentricity
(a) t = 0.06 s., (b) t = 0.12 s., (c) t = 0.09 s., (d) t = 0.15 s.
47
Figure 5.21 Normal components of airgap flux density for mix eccentricity
(a) t = 0.06 s., (b) t = 0.12 s., (c) t = 0.09 s., (d) t = 0.15 s.
48
5.6 Stator Asymmetry
5.6.1 Inter-turn Short Circuit on Stator Windings
The inter-turn short circuit fault can result in some distortion in the stator MMF. This is due to less numbers of the stator windings and the opposite MMF which is against the main MMF as shown in Figure 3.7. To implement this fault in the FEM model, both the geometry and circuit have to be modified.
Consider the stator circuit. According to [53], a coil conductor represents thin wires, in which the induced currents are zero or negligible. The equation for the coil conductor used in the FEM calculation is expressed as shown in Equation 5.2.
( ) ( ) dV t R i tdtφ
= ⋅ + (5. 2)
Where; R = Resistance of the coil = strandn R⋅ = ( )strandn l Sρ⋅ φ = the flux embraced by the assembly of the coil strands strandS = Cross section of a strand = strandF S n⋅ F = Stacking Factor
In the stator circuit, there are two coil conductors per phase. Each coil conductor represents each phase winding in which the direction of the flowing current is taken into account. With the inter-turn short circuit fault, the stator windings require separating into 2 parts as shown in Figure 3.7.
Consider one coil conductor,
11 1 1
( ( ))( ) ( ) ( ) B wB B B w
d n L i tdV t R i t n R i tdt dtφ
= ⋅ + = ⋅ + (5. 3)
Consider 2 series coil conductors,
32
2 3 2 3
2 32 3
( ) ( ) ( ( ) ) ( ( ) )
[( ) ( )] ( ) ( )
BBB B B B
B B wB B w
ddV t V t R i t R i tdt dt
d n n L i tn n R i tdt
φφ+ = ⋅ + + ⋅ +
+= + +
(5. 4)
If 1 2 3( ) ( ) ( )B B BV t V t V t= + , therefore;
1 2 3B B Bn n n= + (5. 5)
Equation 5.5 shows that a coil conductor can be separated to several coil conductors.
The modified stator circuit is shown in Figure 5.22. One set composed of two coil conductors representing the whole phase stator windings can be divided into three sets. The first and second sets; i.e. Set1_A and Set2_A, represent the windings in a half of one slot, in which
49
the flowing current direction is taken into consideration. Set2_A stands for the winding turns, in which the shorted circuit will be implemented. Set1_A describes the rest of windings in the half slot excluding the defected windings. For the third set, Set3 represents the remaining windings in the rest of the slots.
Figure 5.22 Modified stator circuit for the healthy condition
Since the windings in the stator slot are separated to two parts, the geometry has to be divided as well. In this study, a three-inter-turn short circuit fault is modeled. According to the motor specification shown in Table 5.1, the number of conductors in a half of one slot is 15 turns. In order to keep the stacking factor constant for all winding sets, the proper dimensions in the geometry for the short circuit windings are required.
Figure 5.23 shows the cross section of the winding turns in a half of one slot. The defected turns are at 7, 8 and 9. Figure 5.24 shows the modified geometry which has been created with the proper dimensions. Moreover, the circuit parameters have to be also adjusted as shown in Table 5.6.
To implement the three-inter-turn short circuit case, the circuit in Figure 5.22 has to be modified to make a short circuit loop as shown in Figure 5.25.
Figure 5.23 The cross section of the winding coil
50
Figure 5.24 Modified geometry for the inter-turn short circuit
Figure 5.25 Modified stator circuit for 3-inter-turn short stator circuit
In Table 5.7, the simulation results on the instantaneous currents for the modified circuit in Figure 5.22 and the original circuit in Figure 5.4 are presented. In addition, the normal component of the airgap flux density of both circuits is compared in Figure 5.26. The results show clearly that both circuits are compatible.
51
Components Definition Parameter
SS1AP1P1_ADD Coil - A (+), Slot #1, 1N turns
1( )slot shortN N N= +
SS1AM1P1_ADD Coil - A (-), Slot #9, 1N turns
1( )slot shortN N N= +
1N N= turns
1,
,total windings
total windings
NR RN
= ×
Stacking factor = . .S F
Resis1110 End Winding Resistance, A(+) Phase, Slot #1
Resis1111 End Winding Resistance, A(-) Phase, Slot #9
1,
,
12 total end
total windings
NR RN
= ⋅ ×
Induc1106 End Winding Inductance, A(+) Phase, Slot #1
Induc1107 End Winding Inductance, A(-) Phase, Slot #9 1
,,
12 total end
total windings
NL LN
= ⋅ ×
SS1AP1ST1 Coil - A (+), Slot #1, shortN turns, (Defected Turns)
SS1AM1ST1 Coil - A (-), Slot #9, shortN turns, (Defected Turns)
shortN N= turns
,,
shorttotal windings
total windings
NR RN
= ×
Stacking factor = . .S F
Resis1112 End Winding Resistance, A(+) Phase, Slot #1 (Defected Turns)
Resis1113 End Winding Resistance, A(-) Phase, Slot #9 (Defected Turns)
,,
12
shorttotal end
total windings
NR RN
= ⋅ ×
Induc1108 End Winding Inductance, A(+) Phase, Slot #1 (Defected turns)
Induc1109 End Winding Inductance, A(-) Phase, Slot #9 (Defected turns)
,,
12
shorttotal end
total windings
NL LN
= ⋅ ×
SS1AP1P1 Coil - A (+), ,( )total windings slotN N− turns
SS1AM1P1 Coil - A (-), ,( )total windings slotN N− turns
,total windings slotN N N= − turns
,,
,
total windings slottotal windings
total windings
N NR R
N−
= ×
Stacking factor = . .S F
STAR1_1 End Winding Resistance, A Phase ,
,,
total windings slottotal end
total windings
N NR R
N−
= ×
STAL1_1 End Winding Inductance, A Phase ,
,,
total windings slottotal end
total windings
N NL L
N−
= ×
SS1BP1P1 Coil - B (+), ,total windingsN turns
SS1BM1P1 Coil - B (-), ,total windingsN turns
SS1CP1P1 Coil - C (+), ,total windingsN turns
SS1CM1P1 Coil - C (-), ,total windingsN turns
,total windingsN N= turns
,total windingsR R=
Stacking factor = . .S F
STBR1_1 End Winding Resistance, B Phase
STCR1_1 End Winding Resistance, C Phase ,total endR R=
STBL1_1 End Winding Inductance, B Phase
STCL1_1 End Winding Inductance, C Phase ,total endL L=
Table 5.6 Parameters for the modified stator circuit
52
Instantaneous Current [A]
Component Original Circuit Modified Circuit Ua -32.9095 -32.9095 Ub -30.7632 -30.7630 Uc 63.6727 63.6725
Table 5.7 Instantaneous current value in each phase at t = 0.025 second, Healthy condition
0 0.2 0.4 0.6 0.8 1 1.2 1.4-1.5
-1
-0.5
0
0.5
1
1.5
Distance [m]
Flux
Den
sity
[Tes
la]
Normal component of airgap flux density, Healthy condition, slip = 0.01
Original Circuit
Modified Circuit
Figure 5.26 Normal components of airgap flux density for different circuits, Healthy condition
Current (rms) [A] Healthy 3 Inter-turn fault
Ua 44.3616 47.1687 Ub 44.3623 44.2722 Uc 44.3626 45.7056
Table 5.8 Current amplitude (rms) for the healthy and 3 inter-turn short circuit conditions
It can be seen from Figure 5.27 that the normal component of the airgap flux density in the case of the inter-turn short circuit fault is distorted from the healthy one. In addition, the flowing current in the short circuit loop is equal to 465.1288 A (rms), flowing in the opposite direction to the main current. Moreover, it can be observed from Table 5.8 that the three phase currents are unbalanced. This results from the decreased impedance of the A-phase windings, and it causes the A-phase currents is higher that the others. By this, the model for inter-turn short circuit has been verified.
53
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-1.5
-1
-0.5
0
0.5
1
1.5
Distance [m]
Airg
ap F
luxd
ensi
ty [T
esla
]
Normal component of airgap flux density, Slip = 0.01, t=2.5 s., Zeta=[-20,90]
Healthy Condition
3 Interturn fault
Figure 5.27 Normal components of airgap flux density for healthy and 3-inter-turn short circuit conditions
5.7 Conclusion
In this chapter, FEM simulation models for different faults have been presented. For the broken rotor bar, there are two possible approaches. The first one is to modify the rotor circuit by removing the element representing the defected bar, and the material properties of the defected bar requires modifying. The second is to change the value of the rotor bar resistance to be a high value. For the broken end ring, the approaches are similar to those of the broken rotor bar. The approach by removing the component is recommended since the variable in the FEM calculation is reduced, so the calculation time will decrease. However, in this study, the other approach is chosen due to the ease of implementation.
For the static eccentricity, the circuit does not require any modification. Only the geometry requires adjusting. By shifting the stator geometry away from the motor center but keeping the rotor center at the same position with the motor center, the FEM model for static eccentricity can be obtained. The FEM model for the dynamic eccentricity can be performed by shifting the rotor geometry instead of the stator center. For the mix eccentricity, both effects from the static and dynamic eccentricities are implemented in the model. The model can be formed by having the motor center between the stator and rotor centers.
Lastly, for the inter-turn short circuit fault, both the geometry and circuit have to be modified. The portion in the rotor slot geometry is required to represent the defected turns. In addition, the stator circuit needs to be expanded from only a set of coils representing the whole winding, to several sets of coils.
54
6 FEM SIMULATION – MOTOR CURRENT SIGNATURE ANALYSIS
6.1 Introduction
In this and the next two chapters, FEM simulations are implemented to study characteristics of various faults. The three different approaches; i.e. MCSA, IPA and EPVA, are applied to analyze the simulation results; therefore, the theoretical study presented in Chapter 3 and 4 can be compared to the simulation results.
This chapter begins with a study of single fault cases. Firstly, three main fault types with six simulation cases are considered. The first fault is rotor asymmetry caused by 10-broken rotor bars and 4-broken end rings. Next, airgap asymmetry is considered. Three fault types are addressed; 40% static eccentricity, 40% dynamic eccentricity and Mix of 40% static and 40% dynamic eccentricities. Lastly, stator asymmetry is studied by the case with 6-inter-turn short circuit. In addition, some additional cases, which have been played with parameters, such as operating slips, severe levels etc., have been investigated. The details can be found in Appendix III.
Next, consequences of fault combinations are studied. Eight cases, which are combinations of two or three different faults, are considered. Lastly, two abnormal operating conditions, which are unsymmetrical supply voltages and oscillating loads, are investigated.
The content of this chapter begins with the details of the FEM model. Then, MCSA has been implemented. Next, characteristics of each fault are discussed. In Chapter 7 and 8, IPA and EPVA have been applied to the simulation results respectively.
6.2 FEM model
The studied induction motor is HXR400LD6, of which specifications have been presented in the previous chapter. The details shown in Table 6.1 are the simulation parameters. It should be noted that the number of elements and nodes for all the simulations are not constant, but they are not much different for all cases. The geometry and elements of the studied induction motor are shown in Figure 6.1.
The time step for all the simulations is 0.5 milliseconds. This time step is capable enough to catch the fundamental component at 50Hz of the supply voltages by 40 points in 1 period. This can show phenomena, which originate from the fundamental component. According to the theory of Nyquist frequency [55], with the sampling time at 0.5 milliseconds or sampling frequency at 2000Hz, the frequency analysis can be considered up to 1000Hz.
Simulation Parameter Operating Condition
Number of Nodes 42661 Supply Voltage 6000 V Iteration Accuracy 1.00E-03 Frequency 50 Hz Step Time 0.0005 s.
Table 6.1 Simulation parameters
55
Figure 6.1 Motor geometry and meshes
6.3 Single fault
Six different faults categorized into three main groups; i.e. rotor, airgap and stator asymmetry, have been studied. The details of the simulation cases can be seen in Table 6.2.
SimulationCase
Description Slip Time [s]
a Healthy condition 1% 4.5 b 10 Broken rotor bars 1% 4.5 c 4 Broken points on end rings 1% 4.5 d 40% Static eccentricity 1% 4.5 e 40% Dynamic eccentricity 1% 4.5 f Mix of 40% static and 40% dynamic eccentricities 1% 4.5 g 6-inter-turn short circuit
(3 turns in A+ , 3 turns in A-) 1% 4.5
Table 6.2 Descriptions of the simulation cases for various types of faults
Begin with the frequency analysis of one-phase stator current as shown in Figure 6.2. The x axis represents frequencies from 0Hz to 1000Hz, and the y axis represents amplitude in Log scale from 0.0025A. In addition, the amplitude of three phase currents and powers is presented in Table 6.3. Chart 6.1 shows the summary of fault signatures from the theoretical study.
56
Chart 6.1 Current Spectrums for various faults
Current [A] (rms) Power [VA] (rms) Case Ia Ib Ic Pa Pb Pc Remark
a 44.361 44.363 44.362 1.709E+05 1.709E+05 1.709E+05 Balanced b 41.624 41.620 41.624 1.472E+05 1.473E+05 1.472E+05 Balanced c 42.105 42.105 42.109 1.586E+05 1.586E+05 1.586E+05 Balanced d 44.337 44.339 44.343 1.708E+05 1.708E+00 1.708E+05 Balanced e 44.496 44.497 44.502 1.715E+05 1.715E+06 1.715E+05 Balanced f 44.466 44.468 44.473 1.714E+05 1.714E+05 1.714E+05 Balanced g 51.986 45.670 46.148 1.949E+05 1.660E+05 1.793E+05 Unbalanced
Table 6.3 Current and power (rms) for various cases
57
Figure 6.2 Current spectrums for various fault types (Red = expected component, Green = unexpected
component)
(a)
(b)
(c)
(d)
(e)
(f)
(g)
58
Consider healthy condition shown in Figure 6.2(a)
• The main spectrums at the fundamental, 5th, 7th, 11th and 13th harmonics have been found, all of which correspond to the fundamental frequency and the phase-belt harmonics of the supply voltages.
• Some unexpected components at 384Hz, 484Hz, 684Hz, 716Hz, 784Hz, and 928Hz exist.
Consider rotor asymmetry shown in Figure 6.2(b) and (c)
• The influences on the line current from the cases of 10-broken rotor bars and 4-broken rotor end rings look similar.
• Some sidebands around the fundamental, 5th, 7th, 11th and 13th harmonics exist. The simulation results agree well with the theoretical study as shown in Table 6.4.(Some cases played with the operating slips are shown in Appendix III (A) and (B) )
• Some unexpected sidebands exist around the components at 170Hz and 760Hz. Their characteristic is similar to the sidebands around the fundamental.
1f s k 1(1 2 )brbf ks f= m [Hz]
kp 1[( )(1 ) ]brb
kf s s fp
= − ±
[Hz] 50 0.01 1 49 1 50
51 49 2 48 5 248 52 247 3 47 7 347 53 346 4 46 11 545 54 544
Table 6.4 Expected spectrums for rotor asymmetry
Consider static eccentricity shown in figure 6.2(d)
• The particular components at 872Hz and 972Hz have been found. They agree well with the theoretical study as shown in Table 6.5. However, they are not the principle rotor slot harmonics, but they are the influences from the 3rd and 5th time harmonics of the supply voltages.
• Due to the step time, it is not possible to see the principle rotor harmonics. However, the components at 872Hz and 972Hz are evident for the static eccentricity. In addition, their amplitude is proportional to the severe degree. (Some additional cases playing with the operating slips and severity are shown in Appendix III (C) )
• Some unexpected components at 33Hz, 594Hz and 694Hz exist.
59
Consider dynamic eccentricity shown in figure 6.2(e)
• The particular components at 839Hz and 939Hz have been found. These correspond to the rotor slot harmonics considering dynamic eccentric orders as shown in Table 6.5. However, they are not the results from the fundamental component but from the 3rd and 5th time harmonics of the supply voltages.
• As similar as static eccentricity, the amplitude of these components is proportional to the severe degree of dynamic eccentricity. (Some additional cases playing with the operating slips are shown in Appendix III (D) )
• The unexpected component at 627Hz exists.
Consider mix eccentricity shown in Figure 6.2(f)
• The influences from the static and dynamic eccentricities are present by the components at 839Hz, 872Hz, 939Hz and 972Hz. (See Appendix III (E) )
• The results of the amplitude modulation between the influences from the static and dynamic eccentricities take place at 17Hz, 33.5Hz 66.5Hz and 83Hz. This agrees well with the theoretical study.
• Some unexpected components at 233.5Hz/265.5Hz and 333.5Hz/365.5Hz arise around the 5th and 7th harmonics. It is possible to conclude that they result from the amplitude modulation due to their frequency, which corresponds to the rotational frequency. It is also possible to make the expression for these components as Equation 6.1. However, their amplitude is small.
, 1mix ecc rf nf mf= + (6. 1)
where n = time harmonic order of the supply voltages
m = 1,2,3,…
• The results of the amplitude modulation between the static and dynamic eccentricities take place at 855.5Hz and 955.5Hz as shown in Figure 6.3. Besides, it also exists at 610.5Hz, which is a product of the unexpected components at 594Hz and 627Hz.
Figure 6.3 Amplitude modulation of static and dynamic eccentricities
60
Static Ecc. Dynamic Ecc. Mix of Ecc. 1f s rQ
k n ,s eccf
[Hz] k dn n ,d eccf
[Hz] m ,mix eccf
[Hz] 50 0.01 68 1 1 1172 1 2 3 1305 1 66.5
1072 1005 33.5 3 1272 1239 2 83 972 939 17 5 1372 1 2 5 1405 872 905 1339 839
Table 6.5 Expected spectrums for mix eccentricity
Consider Inter-turn short circuit shown in figure 6.2(g)
• The amplitude of three phase currents is unbalanced as shown in Table 6.3.
• The components around the fundamental component at 16.5Hz, 33Hz and 67Hz exist. They are close to the expected ones from the theoretical study as shown in Table 6.6. However, the deviation may result from the unbefitting frequency resolution as called “leakage". (The spectrum does not exist at the exact frequency. The “leakage” phenomenon is that the spreads of energy from a single frequency to many frequency locations because the frequency resolution does not match with the frequency of the spectrum [55].) (Some additional cases playing with the operating slips are shown in Appendix III (F) )
• The 3rd and 9th harmonics become dominant.
• The components at 872Hz and 972Hz, which correspond to the rotor slot harmonics, take place. This agrees well with the theoretical study.
• Some unexpected components at 494Hz, 594Hz, 728Hz, 828Hz and 928Hz arise.
• Between mix eccentricity and the inter-turn short circuit fault, their signature in the current looks quite similar. However, the components influenced from the dynamic eccentricity have been found only in the mix eccentricity. On the other hand, the 3rd and 9th harmonics are the proof of the inter-turn short circuit fault.
1f p s
rQ n
k
stf [eq(3.23)]
[Hz]
k
m
n
stf [eq (3.24)]
[Hz]
k
i
m
n
stf [eq (3.25)]
[Hz]
50 3 0.01 68 1 1 66.5 1 0 -1 1072 0 1 0 1 17 1 1 -33.5 1 0 1 1172 0 2 0 1 33.5 2 1 83 1 1 1 1272 0 3 0 1 50 1 3 -17 1 2 1 1372 0 4 0 1 66.5 1 3 166.5 1 -1 -1 972 0 5 0 1 83 1 5 133.5 1 -2 -1 872
Table 6.6 Expected spectrums for inter-turn short circuit
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6.4 Fault Combinations
In this section, the combinations between rotor, airgap and stator asymmetry have been studied. The details of simulation cases are presented in Table 6.7.
Group 5 Broken rotor bars
40% Static Ecc.
40% Dynamic
Ecc.
Mix of 10% Static & 40% Dynamic Ecc
6-inter-turn short
Slip
Simulation
time [s]
X X 1% 2.5 X X 1% 2.5 A X X 1% 2.5
B X X 1% 2.5 X X 1% 2.5 X X 1% 2.5 C X X 1% 2.5
D X X X 1% 2.5
Table 6.7 Descriptions of the simulation cases for various fault combinations
The simulation of the combination between broken rotor bars and end rings has been studied. The conclusion for this case is that there is no interaction from both faults due to the fact that the phenomena of both faults are similar due to the effect in the rotor current distribution. Therefore, rotor asymmetry with only the broken rotor bar fault has been studied in the fault combinations. (Appendix III (G))
6.4.1 Combination of Rotor and Airgap Asymmetry
The frequency analysis on one phase current for the combination of the rotor and airgap asymmetry is shown in Figure 6.4 and 6.5.
Consider the combination of the broken rotors bar and static eccentric faults shown in Figure 6.4(a) and 6.5(a)
• The influences from the individual faults still exist in the line currents; i.e. Sidebands influenced from the rotor asymmetry placed around the fundamental component, the 5th and 7th harmonics and the rotor slot harmonics at 872Hz and 972Hz. However, it is hard to observe the latter due to their small amplitude.
• The interactions between both faults;
o The components at 17Hz, 33Hz, 66.5Hz and 83Hz become dominant. In addition, the sidebands around the 5th, 7th, 11th, 13th, 17th and 19th arise. Their frequency corresponds to the rotational frequency as similar as the components placed around the fundamental.
o Sidebands, of which frequency corresponds to the twice-slip frequency, exist around the components mentioned above.
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Consider the combination of the broken rotor bar and dynamic eccentric faults shown in Figure 6.4(b) and 6.5(b)
• Similar to the previous case, the influences from the individual faults still exist in the line currents at 839Hz and 939Hz. However, it is difficult to find out which individual fault the components at 839Hz and 939Hz result from since the rotor asymmetry can also cause sidebands existing at these components according to Equation 3.6.
• The interaction between both faults are the sidebands, of which frequency corresponds to the twice slip frequency, around the components influenced from the dynamic eccentricity.
Consider the combination of the broken rotor bar and mix eccentric faults shown in Figure 6.4(c) and 6.5(c)
• Similarly, the influences from the individual faults still exist in the line currents. However, its consequences are similar to that of the combination between the rotor asymmetry and the static eccentricity.
• It is hard to separate these three combination cases and identify which eccentric type is.
Figure 6.4 Current spectrums for the various combinations of rotor and airgap asymmetry
(a)
(b)
(c)
63
Figure 6.5 Zoom FFT of current spectrums for various combinations of rotor and airgap asymmetry
64
6.4.2 Combination of Rotor and Stator Asymmetry
The frequency analysis on one phase current for the combination of the broken rotor bar and inter-turn short circuit faults is shown in Figure 6.6 and 6.7.
• The influences from the individual faults still exist in the line currents; i.e. Sidebands influenced from the rotor asymmetry placed around the fundamental component, the 5th and 7th harmonics and the components at 0.5Hz, 17Hz, 33.5Hz, 66.5 Hz, 83Hz and 99.5 Hz as well as the 3rd and 9th harmonics, which are the influences from the stator asymmetry.
• The interactions between both faults;
o Sidebands influenced from the rotor asymmetry arise around the components influenced from the stator asymmetry.
o In addition, the components influenced from the stator asymmetry also exist not only around the fundamental component but also around the components influenced from the rotor asymmetry according to Equation 3.6 as shown in the table below.
Component Center (Equation 3.6)
116Hz, 132.5Hz, 165.5Hz and 182Hz
149Hz
215Hz, 231.5Hz, 264.5Hz and 281Hz
248Hz
230.5Hz/363.5Hz 347Hz
Figure 6.6 Current spectrums for the combination of rotor and stator asymmetry
65
Figure 6.7 Zoom FFT of current spectrums for the combination of rotor and stator asymmetry
6.4.3 Combination of Airgap and Stator Asymmetry
Consider the combination of the static eccentricity and the inter-turn short circuit fault shown in Figure 6.8
• The influences from the individual faults still exist in the line currents as can be observed by the components at 0.5Hz, 16.5Hz, 33Hz, 66.5 Hz and 83Hz as well as the rotor slot harmonics at 872Hz and 972Hz. Since the consequences of the static eccentricity and the inter-turn short circuit fault are quite similar, it is therefore hard to identify which fault these components result from. However, the dominant 3rd and 9th harmonics, which are the proof of the stator asymmetry, are still present.
• There is no proof from the fault interaction due to the similar phenomena in the line current. Therefore, it is hard to separate this combination from the case with the individual faults as the stator asymmetry or the static eccentricity.
• Some new components at 584Hz, 628Hz, and 838.5Hz arise, but their amplitude is small.
66
Figure 6.8 Current spectrums of the combination of 40% static eccentricity and 6-inter-turn short circuit, compared to the individual faults
Consider the combination of the dynamic eccentricity and the inter-turn short circuit fault shown in Figure 6.9
• Similarly, the influences from the individual faults still exist in the line currents; i.e. the particular components from the dynamic eccentricity at 627Hz, 839Hz and 939Hz and the particular components from the inter-turn short circuit fault at 0.5Hz, 16.5Hz, 33Hz, 66.5 Hz, 728Hz, 772Hz, 828Hz, 872Hz, 928Hz and 972Hz as well as the dominant 3rd and 9th harmonics.
• The interactions between both faults;
o The components at 83Hz, 133.5Hz, 166.5Hz, 183Hz, 233.5Hz, 266.5Hz and 366.5Hz become dominant. These components may result from only the stator asymmetry or from the interaction of both faults.
o The result of the amplitude modulation between the components from the dynamic eccentricity and the inter-turn short circuit fault arises at 955.5Hz, which originate from the components at 939Hz and 972Hz. This component is also found in the mix eccentricity due to the fact that the stator asymmetry also causes the rotor slot harmonics.
67
o The components at 510.5Hz, 711.5Hz, 811.5Hz, 888.5Hz 911.5Hz and 988.5Hz take place. These components may result from the amplitude modulation. It can be observed that they are placed away from the components influenced from the stator asymmetry with the frequency corresponding to the rotational frequency. For example, 510.5Hz (16.5Hz from 494Hz), 711.5Hz (16.5Hz away from 728Hz), 811.5Hz (16.5Hz away from 828Hz), 888.5Hz (16.5Hz away from 872Hz), 911.5Hz (16.5Hz away from 928Hz) and 988.5Hz (16.5Hz away from 972Hz)
• It is hard to separate this fault combination from the individual mix eccentricity due to the similar consequences on the line currents.
Figure 6.9 Current spectrums of the combination of 40% dynamic eccentricity and 6-inter-turn short circuit, compared to the individual faults
Consider the combination of the mix eccentricity and the inter-turn short circuit fault shown in Figure 6.10
• Similarly, the influences from the individual faults still exist in the line currents.
68
• Some new components at 510.5Hz, 711.5Hz, 811.5Hz, 888.5Hz, 911.5Hz and 988.5Hz arise. The components also arise in the combination case between the dynamic eccentricity and the inter-turn short circuit fault.
• Due to the consequences of the mix eccentricity originating from the static and dynamic eccentricities, it is therefore hard to distinguish this combination from the combination of the dynamic eccentricity and the inter-turn short circuit fault.
Figure 6.10 Current spectrums of the combination of mix of 10% static & 40% dynamic eccentricities and 6-inter-turn short circuit, compared to the individual faults
6.4.4 Combination of Rotor, Airgap and Stator Asymmetry
The frequency analysis on one-phase current for the combination of three faults is shown in Figure 6.11 and 6.12.
• The result is similar to that of the combination between the rotor and stator asymmetry. Therefore, it is hard to distinguish this combination from the latter case.
69
Figure 6.11 Current Spectrums for the combination of 5-broken rotor bars, mix of 10% static and 40% dynamic eccentricities and 6-inter-turn short circuit
Figure 6.12 Zoom FFT of current spectrums for the combination of 5-Broken rotor bars, Mix of 10% static and 40% dynamic eccentricities and 6-inter-turn short circuit
6.5 Unsymmetrical Supply Voltage & Oscillating Loads
In this section, two additional cases related to abnormal operating conditions have been studied. The first case is the unsymmetrical supply voltages. The simulation has been performed by increasing 10% of the normal A-phase voltage. The second case is the
70
oscillating load condition. It should be noted that the latter case is studied by an analytical model of induction motors by Simulink. The analytical model is composed of the dynamic model of induction motors and the mechanical coupling model. Its detail is described in Appendix IV. The frequency of the studied oscillating loads is 2Hz as shown in Figure 6.13. The current spectrums of both cases are presented in Figure 6.14.
The studied induction motor for the oscillating loads is different from the previous one. The details are shown in Table 6.9. In addition, it should be noted that the analytical model of induction motors cannot express all phenomena of induction motors. For example, slot harmonics, saturation etc. However, it is good enough for the transient and steady state phenomena.
Motor Specifications Equivalent Circuit Parameters Power 1.8 kW Stator resistance 0.767 Ω Voltage 230 V, 3phase, 50Hz Rotor resistance 1.04 Ω Current 7.8 A. Stator reactance 1.4425 Ω Number of pole 4 poles Rotor reactance 1.4425 Ω Rated Speed 1400 rpm Magnetizing reactance 22.01 Ω Moment of Inertia 50 pu.
Table 6.8 Specifications and equivalent circuit parameters of the studied motor for the load oscillation
Figure 6.13 2Hz Load Oscillation
Consider unsymmetrical supply voltage shown in Figure 6.14(a)
• The three phase stator currents are unbalanced. ( aI = 51. 78 A., bI = 38.66 A., cI = 48.965 A.)
• The 3rd and 9th harmonic components become dominant.
• Compare to the current spectrums of the inter-turn short circuit fault. There is no component corresponding to the rotational frequency around the fundamental in this case.
Consider oscillating load condition shown in Figure 6.14(b)
• With the 2Hz oscillating loads, the sidebands at 46Hz, 48Hz, 52Hz, and 54Hz exist. Their frequency corresponds to the frequency of the oscillating loads. This agrees well with the theoretical study and the experiments in [43].
71
Figure 6.14 Current spectrums for the 10% unsymmetrical supply voltage and the 2Hz oscillating load.
6.6 Conclusion
In summary, the fault signatures in the line current are unique. By considering the particular components in the current spectrums, faults can be identified. The table below is the summary of the fault signatures in the line current.
Case
Unb
alan
ced
Cur
rent
The
3rd
and
9th
harm
onic
s1
Rot
or s
lot
harm
onic
s2
Rot
or s
lot
harm
onic
s co
nsid
erin
g dy
nam
ic o
rder
s2
Side
band
s at
Side
band
s ar
ound
the
fund
amen
tal
Note
Rotor asymmetry X A Static eccentricity X X C Dynamic eccentricity X D Mix eccentricity X X X E Inter-turn short circuit X X X X Unbalanced supply voltage X X Oscillating load X B
(a)
(b)
72
Note: A: The sideband characteristic corresponds to the twice slip frequency, and they can be observed at the 5th and 7th harmonics according to Equation 3.5 and 3.6.
B: The sideband characteristic corresponds to the frequency of load oscillation.
C: The rotor slot harmonic component and the sidebands may not show significantly due to the small degree of static eccentricity.
D: The slot harmonic component may not show significantly due to the small degree of dynamic eccentricity.
E: The sidebands can be observed at the 5th harmonics according to Equation 6.1. However, they may not show significant due to the small degree of eccentricities.
Remark: 1: They originate from the 3rd and 9th harmonics on the supply voltage.
2: it is hard to measure the component due to its small amplitude.
However, it should be noted that there is no standard value for threshold level. Therefore, the diagnosis becomes more reliable if there is some measurement from a healthy condition to be a reference.
For the fault combinations, the influences from individual faults still exist in the line currents. Some new components resulting from the amplitude modulation also arise. To sum up, rotor asymmetry in combination with other faults creates sidebands corresponding to the twice slip frequency 12ksf± around the particular components from the other faults. With stator asymmetry, sidebands corresponding to the rotational frequency rmf± arise. This is valid only with rotor asymmetry. With airgap asymmetry, the influence from stator asymmetry does not show noticeably.
73
7 FEM SIMULATION (CONTINUED) – INSTANTANEOUS POWER APPROACH
7.1 Introduction
This chapter is the continued part from the previous chapter. IPA has been applied to investigate the simulation results. The studied cases are similar to ones in the previous chapter.
7.2 Single fault
With IPA, one phase instantaneous power is used for fault identification instead of one phase current. Frequency analyses on one phase instantaneous power for all the studied fault types are shown in Figure 7.1. The fault signatures for IPA are summarized in Chart 7.1.
Chart 7.1 Instantaneous Power Spectrums for various faults
74
Figure 7.1 Instantaneous power spectrums for various types of faults
(a)
(b)
(c)
(d)
(e)
(f)
(g)
75
Consider healthy condition shown in Figure 7.1(a)
• The main components show at DC, 100Hz, 200Hz, 300Hz and 400Hz. The DC and component at 100Hz result from the fundamental frequency of the supply voltages and currents. The components at 200Hz, 300Hz and 400Hz are from the interactions of the first three harmonics of voltages and currents. These agree well with the theoretical study.
• Some unexpected components at 334Hz, 434Hz, 534Hz, 634Hz, 666Hz and 834Hz arise. They may result from the amplitude modulation between the fundamental component of the supply voltages and the components at 384Hz, 484Hz, 684Hz, 716Hz and 784Hz in the line current (they have been found in the line current as explained in Chapter 6).
Figure 7.2 Amplitude modulation
Consider rotor asymmetry shown in Figure 7.1(b) and (c)
• The influences on the instantaneous power from the case of 10-broken rotor bars and 4-broken rotor end rings look similar.
• The sidebands, which are placed around DC and 100Hz with a particular frequency corresponding to the twice slip frequency 12ksf± as shown in Table 7.1, take place. This agrees well with the theoretical study.
• The advantage of IPA can be seen obviously from the high amplitude level of the sidebands.
• Sidebands also exist around the components at 200Hz, 300Hz,…,1000Hz. This is due to the amplitude modulation between the time harmonics of voltages and currents.
1f s k 12brbf ksf=[Hz]
k 1(2 2 )brbf ks f= m [Hz]
50 0.01 1 1 1 99 2 2 101 3 3 2 98 102 3 97 103
Table 7.1 Expected spectrums for rotor asymmetry
76
Consider static eccentricity shown in Figure 7.1(c)
• The particular components at 544Hz, 822Hz and 922Hz exist. They result from the amplitude modulation between the rotor slot harmonics found in the line currents; i.e. the components at 594Hz, 872Hz and 972Hz, and the fundamental component of the supply voltages.
• It is possible to make the expression for the rotor harmonic components in the instantaneous power as shown in Equation 7.1. The additional term 1± in Equation 7.1 results from the amplitude modulation of the fundamental frequency of the supply voltages.
, 1(1 )[( ) 1]s ecc r
sf kQ n fp−
= ± ± ⋅ (7. 1)
• The sideband corresponding to the component at 33Hz found in the line currents are converted to show around DC and the 100Hz components at 17Hz and 83Hz in the instantaneous power.
Consider dynamic eccentricity shown in Figure 7.1(d)
• Similar to static eccentricity, the components resulting from the amplitude modulation between the rotor harmonic components with the dynamic eccentric orders and the fundamental component of the supply voltages are present at 577Hz, 677Hz, 789Hz, 889Hz and 989Hz. They originate from the components at 627Hz, 839Hz and 939Hz found in the line currents. The expression for the components influenced from the dynamic eccentricity on the instantaneous power can be expressed as Equation 7.2.
, 1(1 )[( ) 1]d ecc r d
sf kQ n n fp−
= ± ± ± ⋅ (7. 2)
Consider Mix eccentricity shown in Figure 7.1(e)
• The sidebands corresponding to the rotational frequency in the line current are converted to arise around DC at 16.5Hz and 33Hz, and around the 100Hz component at 83.5Hz and 116.5Hz. This agrees well with the theoretical study.
• The sidebands corresponding to the rotational frequency also exist around the 200Hz, 300Hz and 400Hz components at 183.5Hz/216.5Hz, 283.5Hz/316.5Hz, and 383.5Hz/ 416.5Hz.
• The influences from the static and dynamic eccentricities as well as the consequences from the amplitude modulation are present. This can be observed by the components at 544Hz /560.5Hz /577Hz, 644Hz/ 660.5Hz /677Hz, 789Hz/ 805.5Hz /822Hz, 889Hz/ 905.5Hz /922Hz, and 978Hz/989Hz.
Consider Inter-turn short turn circuit shown in Figure 7.1(g)
• Similar to rotor asymmetry, the amplitude of the characteristic components is high, and they are converted to place around DC and the twice fundamental component as can be seen from the components at 17Hz, 33.5Hz, 66.5Hz, 83Hz and 117Hz
77
• The components from the rotor slot harmonics according to Equation 7.1 are present at 822Hz and 922Hz.
• The results from the amplitude modulation between the fundamental of the supply voltages and the unexpected components found in the line currents arise at 678Hz, 778Hz and 878Hz.
• The dominant 3rd and 9th harmonics in the line currents influenced from stator asymmetry cannot be observed in the instantaneous power due to the amplitude modulation.
• With IPA, it is hard to separate the case with the individual static or mix eccentricities from the inter-turn short circuit fault. However, they can be separated by determining the amplitude of three phase currents.
7.3 Fault Combinations
In conclusion, the influences from the fault combinations showing on the instantaneous power are similar to that on the line currents. The influences from individual faults as well as the consequences of fault interactions still exist on the instantaneous power.
• The combination of rotor and airgap asymmetry shown in Figure 7.3 and 7.4
o It has been found that there is no obvious component, which can identify the eccentric type.
• The combination of rotor and stator asymmetry shown in Figure 7.5 and 7.6
o A ton of sidebands influenced from the stator asymmetry exist around the 200Hz, 300Hz,…,1000Hz components. In addition, sidebands corresponding to the twice-slip frequency take place around the components influenced from the stator asymmetry.
• The combination of airgap and stator asymmetry shown in Figure 7.7-7.9
o There is no obvious proof that can distinguish this combination case from the case with the individual static, dynamic or mix eccentricities as well as the stator asymmetry.
• The combination of rotor, airgap and stator asymmetry shown in Figure 7.10 and 7.11
o The instantaneous power spectrums for the three-fault combination are similar to that of the combination between the rotor and stator asymmetry.
78
Figure 7.3 Instantaneous power spectrums for the combinations of rotor and airgap asymmetry
79
Figure 7.4 Zoom FFT of instantaneous power spectrums for the combinations of rotor and airgap asymmetry
Figure 7.5 Instantaneous power spectrums for the combination of rotor and stator asymmetry
80
Figure 7.6 Zoom FFT of Instantaneous power spectrums for the combination of rotor and stator asymmetry
81
Figure 7.7 Instantaneous power spectrums of the combination of 40% static eccentricity and 6-inter-turn short circuit, compared to the individual faults
82
Figure 7.8 Instantaneous power spectrums of the combination of 40% dynamic eccentricity and 6-inter-turn short circuit, compared to the individual faults
83
Figure 7.9 Instantaneous power spectrums of the combination of mix of 10% static & 40% static eccentricity and 6-inter-turn short circuit, compared to the individual faults
Figure 7.10 Instantaneous power spectrums for the combination of rotor, airgap and stator asymmetry
84
Figure 7.11 Zoom FFT of instantaneous power spectrums for the combination of rotor, airgap and stator asymmetry
7.4 Unsymmetrical Supply Voltage & Load Oscillation
Consider unsymmetrical supply voltage shown in Figure 7.12(a)
• The three phase powers are unbalanced. ( aP = 1.5592e5 VA., bP = 1.1081e5 VA.,
cP = 1.5634e5 VA.)
• The instantaneous spectrums of the unsymmetrical supply voltage looks similar to that of the healthy condition as shown in Figure 7.1(a). This can imply that IPA cannot separate these two conditions.
Consider oscillating loads shown in Figure 7.12(b)
• Similar to MCSA, the components at 2Hz and 4Hz exist around DC as well as the components at 98Hz, 102Hz, and 104Hz also take place around the 100Hz component. Their frequency corresponds to the oscillating loads and agrees well with the theoretical study.
85
Figure 7.12 Instantaneous power spectrums for the 10% unsymmetrical supply voltage and the 2Hz oscillating load
7.5 Conclusion
In summary, IPA is based on MCSA. The components, which exist in the current, also exist in the instantaneous power regarding the results of the amplitude modulation between the fundamental component of the supply voltages and the particular components in the line currents. By this, IPA can gain the benefit that is to convert sidebands placed around the fundamental component in the line current to place around DC and the twice-fundamental frequency component. In addition, amplitude of characteristic components in the instantaneous power is also higher than that in the line current. Hence, characteristic components can be detected easily due to the ease of filtering out DC.
However, there are some drawbacks of IPA. First, IPA is not capable to distinguish static or mix eccentricities from the inter-turn short circuit, and it cannot separate the unsymmetrical supply voltage condition from the healthy condition either. However, this drawback can be managed if the information of three phase rms currents or powers is available. The summary of the fault signatures of IPA is shown in the table below.
(a)
(b)
86
Case
Unb
alan
ced
Pow
er
Com
pone
nts
follo
win
g to
Eq
. 7.1
Com
pone
nts
follo
win
g to
Eq
. 7.2
Side
band
s
aro
und
DC
and
Side
band
s
arou
nd D
C
and
Note
Rotor asymmetry X A Static eccentricity X X C Dynamic eccentricity X D Mix eccentricity X X X E Inter-turn short circuit X X X Unbalanced supply voltage X Oscillating load X B
Note: A: The sideband characteristics correspond to the twice slip frequency, and they can exist around the components at 1 1 14 , 6 ,..., 20f f f as well.
B: The sideband characteristics correspond to the frequency of load oscillation.
C: The components following to Equation 7.1 and the sidebands may not show significantly due to the small degree of static eccentricity.
D: The components following to Equation 7.2 may not show significantly due to the small degree of dynamic eccentricity.
E: The sidebands can exist around the components at 1 1 14 ,6 , and 8f f f . In addition, the
products of the amplitude modulation corresponding to the characteristic frequency rmf ,
can be observed between the components following to Equation 7.1 and 7.2.
For the fault combinations, it is possible to make a similar conclusion as that of MCSA. By IPA, only the combination cases of rotor and airgap asymmetry, rotor and stator asymmetry, and rotor, airgap and stator asymmetry can be identified. However, IPA cannot identify which eccentric type in the combinations is.
87
8 FEM SIMULATION (CONTINUED) – THE EXTENDED PARK’S VECTOR APPROACH
8.1 Introduction
This chapter is the continued part from Chapter 6. EPVA has been implemented to investigate the simulation results. The studied cases are similar to the ones in the previous chapter.
8.2 Single fault
By converting from the three-phase system to the two-phase system, two different indicators can be obtained. The first indicator is the Lissajou’s curve. The deviation of Lissajou’s curve from the healthy one can tell whether an induction motor is in an abnormal condition or not. To identify a fault type, the modulus current is however required. In Chart 8.1, the fault signatures for EPVA are summarized in Chart 8.1.
Chart 8.1 Extended Park’s Vector Approach for various faults
The Lissajou’s curves and current modulus spectrums for each fault type are presented in Figure 8.1-8.4.
88
Figure 8.1 Lissajou’s curves for various fault types
(a) (b)
(c) (d)
(e) (f)
(g)
89
Figure 8.2 Current modulus spectrums for various fault types
(a)
(b)
(c)
(d)
(e)
(f)
(g)
90
Figu
re 8
.3 C
urre
nt m
odul
us w
avef
orm
s fo
r var
ious
faul
t typ
es
(g)
(f)
(e)
(d)
(c)
(b)
(a)
91
Figu
re 8
.4 Z
oom
cur
rent
mod
ulus
wav
efor
ms
for v
ario
us fa
ult t
ypes
(a)
(d)
(e)
(f)
(g)
92
Consider healthy condition shown in Figure 8.1(a), 8.2(a), 8.3(a) and 8.4(a)
• The shape of Lissajou’s curve is circular.
• The DC component shows apparently. This agrees well with the theoretical study. However, the component at 300Hz shows significantly as well. This component results from the amplitude modulation between the fundamental and 5th harmonic of the line currents.
• Some components at 200Hz, 434Hz, 500Hz, 600Hz, 666Hz, 734Hz and 900Hz also exist. They also result from the amplitude modulation of the fundamental and the unexpected components found in the line currents.
Consider rotor asymmetry shown in Figure 8.1(b) and(c), 8.2(a) and (b), and 8.3(a) and (b)
• The influences from the case with 10-broken rotor bars and 4-broken rotor end rings on both the Lissajou’s curve and the current modulus look similar.
• The shape of Lissajou’s curve is still circular, but the rim of the circle becomes thicker. This agrees well with the theoretical study. In addition, it can imply that the output power or torque from an induction motor with rotor asymmetry is oscillating.
• Similar to MCSA and IPA, sidebands around the DC, 300Hz, 600Hz and 900Hz components also exist in the current modulus. Their frequency corresponds to the twice slip frequency 12ksf± .
• The unexpected component at 677Hz arises.
Consider static eccentricity shown in Figure 8.1(d), 8.2(d), 8.3(d) and 8.4(d)
• The Lissajou’s curve does not show any difference from the healthy one. Therefore, the Lissajou’s curve cannot be an indicator of the static eccentricity.
• The component resulting from the amplitude modulation between the fundamental and rotor slot harmonics in the line currents arises at 922Hz. The expression on Equation 7.1 can be also applied for EPVA.
• The particular component at 17Hz exists. This corresponds to the component found in MCSA and IPA.
• The particular component at 644Hz exists. This is a product of the amplitude modulation between the fundamental and the unexpected component at 694Hz found in the line current.
Consider dynamic eccentricity shown in Figure 8.1(e), 8.2(e), 8.3(e) and 8.4(e)
• Similar to the static eccentricity, the Lissajou’s curve cannot be an indicator of the dynamic eccentricity either.
• The components resulting from the amplitude modulation can be found at 677Hz and 889Hz. These are the influences from the components at 627Hz and 939Hz found in the line current. The expression on Equation 7.2 can be also applied for EPVA.
93
Consider Mix eccentricity shown in Figure 8.1(f), 8.2(f), 8.3(f) and 8.4(f)
• Similarly, the Lissajou’s curve cannot be an indicator of the mix eccentricity.
• The sidebands corresponding to the rotational frequency have been found at 16.5Hz and 33Hz. This agrees well with the theoretical study. Not only sidebands around DC, sidebands around the 300Hz component have been found at 283.5Hz and 316.5Hz.
• The particular components influenced from static and dynamic eccentricities have been found at 644Hz, 889Hz and 677Hz, 922Hz respectively. In addition, the products of the amplitude modulation are also found at 660.5Hz and 905.5Hz, which originate from the unexpected components found in the line currents in the case of static and dynamic eccentricities.
Consider Inter-turn short turn shown in Figure 8.1(g), 8.2(g), 8.3(g) and 8.4(g)
• The shape of Lissajou’s curve is obviously elliptical. This is apparently different from healthy and rotor asymmetry ones. Therefore, the Lissajou’s curve can be an indicator of the inter-turn short circuit fault.
• The component at 100Hz becomes dominant. This agrees well with the theoretical study. Besides, the components at 200Hz and 400 Hz also take place. They are caused by the amplitude modulation between the fundamental and 3rd harmonic, and between the 3rd and 5th harmonics in the line currents.
• The sidebands corresponding to the rotational frequency are present at 17Hz, 33.5Hz. In addition, the sidebands around 100Hz and 200Hz also exist at 66.5Hz, 83Hz, 117Hz, 133.5Hz, 166.5Hz and 183.5Hz.
• The components resulting from the rotor slot harmonics arise at 778Hz/ 822Hz, 878Hz/922Hz and 978Hz.
8.3 Fault Combinations
8.3.1 Combination of Rotor and Airgap Asymmetry
The Lissajou’s curve and current modulus are shown in Figure 8.5- 8.7
• Only the influences from rotor asymmetry show apparently by the rim of the Lissajou’s curve, which becomes thicker.
• The influences from the individual faults still exist in the current modulus as can be observed by the sidebands, of which frequency corresponds to the twice slip frequency 12ksf± , and the components, of which frequency corresponds to the rotational frequency. All of them exist around DC and the 300Hz component.
• The consequences of both faults in the current modulus are similar to that of MCSA and IPA. Sidebands influenced from the rotor asymmetry take place around the components influenced from the airgap asymmetry.
• Similar to MCSA and IPA, there is no proof that can identify which eccentric type in the fault combination is.
94
Figure 8.5 Lissajou’s curves for the combination of rotor and airgap asymmetry
95
Figure 8.6 Current modulus spectrums for the combination of rotor and airgap asymmetry
Figure 8.7 Current modulus waveforms for the combination of rotor and airgap asymmetry
96
8.3.2 Combination of Rotor and Stator Asymmetry
The Lissajou’s curve and current modulus are shown in Figure 8.8-8.10
• The influences from the rotor and stator asymmetry can be observed clearly by the thicker and elliptical shape of the Lissajou’s curve.
• The influences from the individual faults still exist in the current modulus as can be observed by the sidebands, of which frequency corresponds to the twice slip frequency 12ksf± and the component, of which frequency corresponds to the rotational frequency. In addition, the component at 100Hz, which is the proof of the stator asymmetry, is present.
• The interactions of both faults in the current modulus are similar to that of MCSA and IPA. Sidebands influenced from the rotor asymmetry take place around the components resulting from the stator asymmetry. In addition, tons of sidebands corresponding to the rotational frequency also exist around the components at 100Hz, 200Hz,…., 1000Hz.
Figure 8.9 Current modulus spectrums for the combination of rotor and stator asymmetry
Figure 8.8 Lissajou’s curve for the combination of rotor and stator asymmetry
97
Figure 8.10 Current modulus waveform for the combination of rotor and stator asymmetry
8.3.3 Combination of Airgap and Stator Asymmetry
Consider static eccentricity and inter-turn short circuit shown in Figure 8.11(a), 8.12, and 8.15(a)
• The influences from the stator asymmetry can be observed clearly by the elliptical shape of Lissajou’s curve.
• The influences from the individual faults still exist in the current modulus
Figure 8.11 Lissajou’s curves for the combinations of airgap and stator asymmetry
(a) (b)
(c)
98
• There is no proof from the interaction of both faults in the current modulus since the stator asymmetry can result in the rotor slot harmonics in the line current. Therefore, it is hard to separate this combination from the case with individual stator asymmetry.
Figure 8.12 Current modulus spectrums of the combination of 40% static eccentricity and 6-inter-turn short circuit, compared to the individual faults
Consider dynamic eccentricity and inter-turn short circuit shown in Figure 8.11(b), 8.13, and 8.15(b)
• Similarly, the Lissajou’s curve can detect only the stator asymmetry but cannot identify dynamic eccentricity. In addition, the influences from the individual faults still exist in the current modulus.
• The consequences of both faults do not show obviously. Some components corresponding to the rotational frequency arise at 216.5Hz, 283.5Hz and 316.5Hz. They may be the influences from the stator asymmetry. Besides, some sidebands corresponding to the rotational frequency arise around the rotor slot harmonics as can be observed by the components at 661.5Hz (around 678Hz), 761.5Hz (around 778Hz), 905.5Hz / 938.5Hz (around 922Hz) and 961.5Hz/994.5Hz (around 978Hz). They also result from the amplitude modulation.
99
• There is no proof in the current modulus which can separate this combination from the case with the individual stator asymmetry or mix eccentricity.
Figure 8.13 Current modulus spectrums of the combination of 40% dynamic eccentricity and 6-inter-turn short circuit, compared to the individual faults
Consider mix eccentricity and inter-turn short circuit shown in Figure 8.11(c), 8.14, and 8.15(c)
• The result is similar to that of the combination between the dynamic eccentricity and stator asymmetry. Therefore, it is hard to distinguish this combination from the latter case.
100
Figure 8.14 Current modulus spectrums of the combination of mix of 10% static and 40% dynamic eccentricities and 6-inter-turn short circuit, compared to the individual faults
Figure 8.15 Current modulus waveforms for the combinations of airgap and stator asymmetry
(a)
(b)
(c)
101
8.3.4 Combination of Rotor, Airgap and Stator asymmetry
The Lissajou’s curve and current modulus are shown in Figure 8.16-8.18
• The patterns found in the Lissajou’s curve and the current modulus of the three fault combination are similar to that of the combination between the rotor and stator asymmetry. It is therefore hard to separate this combination from the combination between rotor and stator asymmetry.
Figure 8.17 Current modulus spectrums for the combination of rotor, airgap and stator asymmetry
Figure 8.18 Current modulus waveform for the combination of rotor, airgap and stator asymmetry
Figure 8.16 Lissajou’s curve for the combination of rotor, airgap and stator asymmetry
102
8.4 Unsymmetrical Supply Voltage & Load Oscillation
Consider unsymmetrical supply voltage shown in Figure 8.19(a), 8.20(a) and 8.21(a)
• The shape of Lissajou’s curve is as elliptical as that of the stator asymmetry. This is because of the negative sequence due to the unsymmetrical supply voltages.
• The component at 100Hz still exists. However, there is no sideband corresponding to the rotational frequency. This can separate this condition from the stator asymmetry.
Consider load oscillation shown in Figure 9.19(b), 8.20(b) and 8.21(b)
• The thicker rim of the Lissajou’s curve can be observed. This is due to the load variation resulting in the fluctuation of the operating slip.
• The sidebands corresponding to the frequency of oscillating loads take place around DC. It should be noted that to recognize the case with rotor asymmetry and the oscillating load condition, information of the operating conditions is required.
Figure 8.19 Lissajou’s curves for the 10% unsymmetrical supply voltage and the 2Hz load oscillation.
(a) (b)
103
Figure 8.20 Current modulus spectrums for the 10% unsymmetrical supply voltage and the 2Hz oscillating load.
Figure 8.21 Current modulus waveforms for the 10% unsymmetrical supply voltage and the 2Hz oscillating load
8.5 Conclusion
In summary, EPVA is based on MCSA. The indicators found in the line currents are also visible in the current modulus. One benefit from EPVA is that an abnormal condition can be easily detected by the deviation of Lissajou’s curve. Similar to IPA, EPVA can convert sidebands around the fundamental component in the line currents to place around DC in the current modulus. However, the drawback of EPVA is that it requires more measurement than MCSA and IPA.
From the simulation results presented above, the fault signatures for EPVA can be summarized as shown in the table below. As similar to MCSA and IPA, it is hard for EPVA to detect static and dynamic eccentricities due to the small influences in the line currents, but EPVA is still effective to detect mix eccentricity.
(a)
(b)
(a)
(b)
104
Lissajou's Curve
Case
Thic
ker
Ellip
tical
Com
pone
nts
follo
win
g to
Eq.
7.1
Com
pone
nts
follo
win
g to
Eq.
7.2
Spe
ctru
m a
t
and
side
band
s
arou
nd D
C
Sid
eban
d ar
ound
D
C
Note
Rotor asymmetry X X A Static eccentricity X X C Dynamic eccentricity X D Mix eccentricity X X X E Inter-turn short circuit X X X X F Unbalanced supply voltage X X Oscillating load X X B
Note: A: The sideband characteristics correspond to the twice slip frequency, and they can exist around the components at 1 1 16 , 12 , and 18f f f .
B: The sideband characteristics correspond to the frequency of load oscillation.
C: The components following Equation 7.1 and the sidebands may not show significantly due to the small degree of static eccentricity.
D: The component following Equation 7.2 may not show significantly due to the small degree of dynamic eccentricity.
E: The sidebands can exist around the component at 16 f . In addition, the products of the
amplitude modulation corresponding to the characteristic frequency rmf , can be observed
between the components following to Equation 7.1 and 7.2.
F: The sidebands can also exist around the component at 14 f .
For the combination of 2 or 3 faults, it is possible to make a similar conclusion as that of MCSA and IPA.
8.6 Evaluation for MCSA, IPA and EPVA
According to Chapter 6, 7 and 8, pros and cons of MCSA, IPA and EPVA are compared in Table 8.1. MCSA is simple since only one measurement on one phase current is required. However, it is sometimes hard to search for sidebands that are close to the fundamental component because it is difficult to filter out the fundamental component without affecting adjacent components, and it is also hard to detect components, of which amplitude is small.
EPVA offers an easy and quick method to the determination of an abnormal condition by a Lissajou’s curve. In addition, EPVA converts sidebands to place around DC in the current modulus. By this, detection of sidebands by EPVA; i.e. sidebands from rotor asymmetry with low operating slip, is more effective than by MCSA. However, it requires more information than MCSA and IPA.
The main benefit of IPA is the easy determination of sidebands even they are close to the fundamental component. In addition, the higher amplitude level is another benefit. This can facilitate detection of strange components. Moreover, some additional information from the supply voltages is obtained in the instantaneous power.
105
The recommendation regarding which method to use depends on two aspects. Firstly, IPA is suitable in case reference measurement is not available. Secondly, EPVA is recommended in case that there is reference measurement in a normal condition. EPVA can be implemented on an automatic system by measuring real-time data and comparing the Lissajou’s curve with the history one, so one can tell whether an induction motor is in a normal condition or not.
Pros Cons
MCSA
- Require only 1 phase current
- Hard to detect sidebands close to the fundamental
- Low amplitude - Require three phase rms current for
separating Mix eccentricity and inter-turn short circuit
IPA
- Easy to detect sidebands by filtering out DC
- High amplitude - Contain information from voltages
- Require one phase current and voltage - Require three phase rms power for
separating Mix eccentricity and inter-turn short circuit
EPVA
- Easy to detect an abnormal condition by Lissajou's curve
- Easy to detect sidebands by filtering out DC
- Require three phase currents - Low amplitude - Require a Lissajou's curve of an induction
motor in a normal condition
Table 8.1 Benefits and Drawbacks for MCSA, IPA and EPVA
106
9 IMPLEMENTATION
9.1 Introduction
In the previous chapters, fault signatures as well as the implementation of MCSA, IPA and EPVA have been presented. In this chapter, the conclusions from the previous chapters are utilized to investigate results from a real measurement. The purpose here is to find out possible root causes of the inexplicable sidebands by applying the theoretical study and FEM simulations gained from the previous chapters.
The analysis begins with checking whether the data has been measured correctly or not. The amplitude of the three phase voltages and currents and their phase shifts have been investigated. Then, the raw measurement has been prepared before investigation. The number of measurement data has to be considered in order to obtain a suitable frequency resolution. Then, IPA has been implemented to the measurement.
9.2 Raw Measurement Data
The specification of the investigated induction motor and the details of the measurement are shown in Table 9.1 and 9.2 respectively.
In addition, the vibration signals in the horizontal and vertical directions at the drive-end side have been measured. By considering the frequency analysis in the vibration signals, the motor speed has been estimated. The most dominant peak has been found at 9.976Hz. This corresponds to the rotational speed at 598.92 rpm or 0.3% operating slip when the fundamental frequency is 50.03Hz. (This information is given by Sture Exison, Senior Specialist Drives and Process Systems Automation Teknik och Service, ABB Automation Technologies, Västerås, Sweden)
Power 355 kW Speed 594 rpm Voltage 6000 V Number of stator pole 90 Rated current 47 A Number of rotor slots 85 Power factor 0.76 Connection Star
Table 9.1 Specifications of the investigated induction motor
Data , ,a b cV V V
,a cI I
Sampling time 270 sμ No. of samples 340578
Table 9.2 Details of the measurement data
Figure 9.1 Investigation procedures
107
9.3 Simple Investigation
It is necessary to verify first whether the measurement has been measured correctly or not. The three concerned points in this simple investigation are considered;
1. The phase shift; Since it is possible that indexes or labels of the measurement have been marked incorrectly, it is therefore necessary to check the measurement by considering the phase shift between each phase voltage and between phase voltages and currents.
2. The voltage amplitude; It is necessary to get information about whether the 3-phase voltages are balanced or not
3. The current amplitude; It is necessary to get information about whether the 3-phase currents are balanced or not
Since the induction motor does not have the neutral connection, the current of the remaining phase can be calculated according to Kirchhoff’s current law as shown in Equation 9.1.
0a b ci i i+ + = (9. 1)
Then, the phase shifts and amplitudes of voltages and currents were measured. The details of the investigation are presented in Table 9.3.
Phase Shift
a bV V− b cV V− a aV I− b bV I− c cV I− b cV I− c bV I−
120o Leading
120o Leading
63o Leading
185.4o Leading
57.6o Lagging
63o Leading
63o Leading
Voltage [Volt] Current [A] A B C A B C
3645.13 3630.89 3628.49 35.09 34.46 34.79
Table 9.3 Simple investigation on the measurement data
It can be seen from Table 9.3 that the C-phase voltage lags the C-phase current. This is incorrect since induction motors are an inductive load. In addition, the phase shift between the B-phase voltage and current is different to the A-phase one. However, it has been found that the phase shift between the B-phase voltage and the C-phase current corresponds to the A-phase one. Hence, it can be concluded that the index is incorrect. The C-phase current should be replaced by the B-phase current.
Consider the amplitudes of the three phase voltages. They are a bit different, but it is possible to consider them balanced. The amplitudes of the three phase currents are also a bit unbalanced. However, they should be remarked since they possibly result from the unbalanced supply voltages or influences from faults.
In addition, the current amplitudes are quite small relative to the rated ones. This corresponds to the estimated operating slip, which is really close to zero. On the other word, the motor load is very low.
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9.4 Data Preparation
9.4.1 Number of Sample
Frequency resolution is related to the number of samples, and the maximum analyzed frequency corresponds to the sampling frequency according to the theory of Nyquist frequency as shown in Equation 9.2.
max
1 1
12 2
sampling
total sampling
sampling
sampling
ff
T n T n
ff
T
Δ = = =⋅
= =⋅
(9. 2)
Where, fΔ = frequency resolution n = Number of samples samplingT = Sampling time
samplingf = Sampling frequency
maxf = The maximum frequency of FFT
Therefore, the parameters according to Equation 9.2 should be selected intelligently, so the proper frequency resolution on FFT can be obtained; otherwise, the leakage phenomena will take place. Due to the estimated operating slip at 0.3%, the frequency resolution equal to 0.02Hz has been selected since it can catch the 0.30Hz sidebands influenced from the rotor asymmetry with 15 points.
With the sampling time at 270 sμ , the number of samples is 185,185.182 samples for the 0.02Hz frequency resolution regarding Equation 9.2. The number of samples equal to 185,185 samples has been selected. However, since the number of samples is not an integer, the leakage phenomena may happen.
From the total number of samples, the raw data has been divided into two groups. The first data group is chosen from the first sample to the 185,185th sample, and the second data group is from the 155,394th sample to the last one.
9.5 Investigation
IPA has been implemented to the prepared data. The instantaneous power has been calculated from a product of the A-phase current and voltage. Figure 9.1 shows the instantaneous power of the investigated induction motor. The DC levels of both data groups are 51077.68VA and 66483.33VA respectively. Figure 9.2 to 9.5 show the zoom FFT in the instantaneous power, of which DC has already been filtered out.
The investigation has been carried out based on two aspects. The first one is regarding the estimated operating slip 0.3%. The second regards the operating slip estimated from the rotor slot harmonics.
I. Investigation regarding the estimated operating slip 0.3%
• It can be seen in Figure 9.4 that the dominant components are placed at 0.02Hz, 2.64Hz, 5.32Hz and 7.96Hz. Their frequency does not correspond to the twice slip frequency or the rotational frequency, which are 0.3Hz and 9.976Hz respectively.
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Hence, these components are not the results from rotor asymmetry, mix eccentricity or stator asymmetry.
• The components influenced from rotor asymmetry around the dominant components have been investigated as shown in Figure 9.5. For example, 0.32Hz and 0.62Hz (around 0.02Hz), 2.34Hz and 2.94Hz (around2.64Hz), 5Hz and 5.6Hz (around 5.3Hz). Only the 5Hz component shows significantly in the data group 1, but it cannot be observed in the data group 2. Therefore, it should not result from rotor asymmetry. Thus, it is possible to conclude that there is no defect in the rotor.
• In Figure 9.6, the components, which correspond to the rotor slot harmonics in Equation 7.1 and 7.2 as shown in Table 9.4, could not be found. This has been also confirmed by the frequency analysis of the A-phase current as shown in Figure 9.7 and Table 9.5. Therefore, it is possible to conclude that there is no influence from static or dynamic eccentricities. However, there might be some small degree of them.
1f rQ p s n , 1(1 )[( ) 1]s ecc r
sf kQ n fp−
= ± ± ⋅dn n , 1
(1 )[( ) 1]d ecc r dsf kQ n n f
p−
= ± ± ± ⋅
50.03 85 5 0.003 1 948.02 1 1 957.99 847.96 857.93 747.90 757.87 938.04 837.98 737.92
Table 9.4 Expected frequencies for the rotor slot harmonics in the instantaneous power
1f rQ p s n , 1(1 )[( ) ]s ecc r
sf kQ n fp−
= ± ⋅
dn n , 1(1 )[( ) ]d ecc r d
sf kQ n n fp−
= ± ± ⋅
50.03 85 5 0.003 1 897.99 1 1 907.96 797.93 807.90 888.01 787.95
Table 9.5 Expected frequencies for the rotor slot harmonics in the current
II. Investigation regarding the operating slip estimated from the rotor slot harmonics.
Table 9.6 shows the operating slips estimated from the dominant spectrums as shown in Figure 9.7. It should be noted that the frequency range from 750Hz to 950Hz is equivalent to the operating slips between 0% and 4%.
• The rotor slot harmonics corresponding to the dominant sideband at 2.64Hz could not be observed. This implied that the motor did not operate at the operating slip at 2.64%.
• However, it should be remarked that the rotor slot harmonics at the operating slip 0.3% could not be observed in the line current either.
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In conclusion, the influences from rotor, airgap and stator asymmetry could not be observed in the measurement. Based on the FEM simulation results presented in the previous chapters, one possible root cause of the inexplicable sidebands can be a result from oscillating loads.
Besides, there are other possible causes. One possible cause is the influences from mechanical couplings. For example, influences from gearbox or vibrations from connected mechanical equipment. Another is to have some degree of eccentricity, of which the revolving speed of the rotor around the motor center corresponds to the sideband frequency.
In addition, the reason for the unbalanced currents is possibly caused by the unbalanced voltages.
Data Set 1
Slip Slip (equation 3.14) ( 1)n = (equation 3.18) ( 1, 1)dn n= = f [Hz]
( )n− ( )n+ (- , - )dn n (- , )dn n+ ( , )dn n+ − ( , )dn n+ +
812.96 10.29% 9.23% 11.34% 844.02 6.64% 5.53% 7.73% 850.44 5.89% 4.78% 6.98% 854.78 5.38% 4.25% 6.48% 873.24 3.21% 2.06% 4.34% 888.44 1.42% 0.25% 2.57% 903.36 0.84% 911.70 933.50
(a) Data group 1
Data Set 2
Slip Slip (equation 3.14) ( 1)n = (equation 3.18) ( 1, 1)dn n= = f [Hz]
( )n− ( )n+ (- , - )dn n (- , )dn n+ ( , )dn n+ − ( , )dn n+ +
812.96 10.29% 9.23% 11.34% 847.70 6.21% 5.10% 7.30% 850.36 5.90% 4.78% 6.99% 873.12 3.22% 2.07% 4.35% 911.70 930.32 933.50
(b) Data group 2
Table 9.6 Operating slip estimated from the dominant components around 750Hz-950Hz
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Figure 9.2 A-phase instantaneous Power
Figure 9.3 Zoom FFT of the instantaneous power spectrums between 0Hz to 200Hz
112
Figure 9.4 Zoom FFT of the instantaneous power spectrums between -5Hz to 15Hz
Figure 9.5 Zoom FFT of the instantaneous power spectrums between -0.2Hz to 3Hz
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Figure 9.6 Zoom FFT of the instantaneous power spectrums between 710Hz to 1000Hz
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Figure 9.7 Zoom FFT of the A-phase current spectrums between 750Hz-950Hz
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10 CONCLUSION AND FUTURE WORKS
10.1 Conclusion
This thesis presents the implementation of FEM simulations in the fault diagnostic application for induction motors and the evaluation of three different diagnostic approaches based on measurement from the stator side; i.e. Motor Current Signature Analysis (MCSA), Instantaneous Power Approach (IPA) and The Extended Park’s Vector (EPVA). The characteristics, consequences and detection of the faults by the stator currents are summarized as table below; Fault/Condition Characteristics Consequences Detection -MCSA Broken Rotor Bars or Broken Rotor End Rings
• Unsymmetrical rotor current distribution leads to unsymmetrical rotor MMF
• Backward rotor MMF
• Sidebands corresponding to the twice slip frequency around the fundamental, 5th and 7th harmonic components
Static eccentricity
• Unsymmetrical airgap with space-fixed minimum airgap length
• Steady unbalanced magnetic pull
• Rotor Slot Harmonics
Dynamic eccentricity
• Unsymmetrical airgap with periodical rotating minimum radial airgap length
• Unbalanced magnetic pull
• Rotor Slot Harmonics considered the dynamic eccentric orders
Mix of Static and Dynamic Eccentricities
• Unsymmetrical airgap with periodical rotating minimum radial airgap length
• Unbalanced magnetic pull
• Rotor Slot Harmonics • Rotor Slot Harmonics
considered the dynamic eccentric orders
• Sideband corresponding to the rotor frequency
Inter-turn Short Circuit
• Less number of winding turns and additional opposite stator MMF
• Unsymmetrical stator MMF and windings
• Rotor Slot Harmonics • Sideband corresponding to
the rotor frequency • The 3rd harmonic
component • Unbalanced three phase
currents
Unsymmetrical Supply Voltage
- - • The 3rd harmonic component
• Unbalanced three phase currents
Oscillating Load - - • Sideband corresponding to
the frequency of load oscillation
In addition, the consequences of fault combinations in the stator currents have been investigated. In additional to influences from individual faults, the consequences of fault
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interaction regarding the results of the amplitude modulation between the components of individual faults also take place. However, the consequences from the combination between airgap and stator asymmetry are hard to distinguish due to their similar influences on the stator currents.
The evaluation has been given in two aspects. IPA is suitable if there is no reference measurement. The main feature of IPA is that it can transform the sideband components in the stator current to exist around DC in the instantaneous power. This results from the amplitude modulation between the fundamental component of the supply voltages and the particular components in the stator currents. This feature gives a benefit to make a diagnosis easier due to the ease of filtering DC. Besides, IPA is more suitable for detection of small particular components since power amplitude is normally much higher than current amplitude. However, IPA requires one additional measurement of a phase voltage.
EPVA is recommended if some reference measurement of an induction motor operating in a normal condition is available. The main benefit of EPVA is that it has the Lissajou’s curve, of which the appearance can notify whether a motor operates in an abnormal condition or not. Besides, the modulus current is another indicator, of which feature is similar to that of IPA. EPVA can transform the consequent sidebands in the stator current to be placed around DC in the current modulus. To identify the fault type, it is necessary to determine both indicators. Nevertheless, EPVA requires two more phase current measurement.
Finally, IPA has been applied to the real measurement. Based on the studied FEM simulations, the possible cause for the inexplicable sidebands found in previous analysis is likely the effect from oscillating loads, since no other fault can generate such harmonics.
10.2 Future Works
To verify the given possible cause of the inexplicable sidebands, some more measurements in load varying conditions are required, since they can verify the given possible cause.
In addition, studies on motor loads and mechanical couplings are interesting. Besides, some faults beyond this study are attractive. For example, eccentricities, combination of abnormal operating condition with faults etc.
Furthermore, the other frequency analysis technique called Wavelet is interesting since it is suitable for non-stationary sinusoidal quantities. Thus, it can be possible used to detect faults in the condition of non-constant operating slips. Besides, implementation of statistic techniques, for example, Time-Series Data Mining [19, 30], Neural Network etc., is also fascinating.
Moreover, in reality, the speed of an induction motor is not constant due to effects from mechanical coupling systems. Therefore, implementation of mechanical models in FEM simulations is interesting. With FLUX-to-Simulink Technology in FLUX2D, it may be possible to implement mechanical models by co-simulation with Simulink.
Lastly, implementation of electrical coupling systems in FEM simulations, for example an inverter feeding power to a motor, is interesting. By this, some new approaches; for example, injecting low frequency signal [39] etc, can be applied, or influences from a PWM supply source can be studied. This can be also possibly reached by co-simulation with Simulink.
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LIST OF SYMBOLS
MCSA Motor Current Signature Analysis IPA Instantaneous Power Approach EPVA The Extended Park’s Vector FEM Finite Element Method MMF Megnetomotive Force F Megnetomotive Force EMF Electromotive Force B Airgap Flux Density [Tesla] V Voltage [Volt] I Current [A]
1f Electrical stator current frequency [Hz]
2f Electrical rotor current frequency [Hz]
rf Rotor Frequency [Hz]
1ω Electrical angular speed of supply voltage [rad/s]
rω Electrical rotor angular speed [rad/s]
mω Mechanical rotational speed [rad/s] p Number of pole pair q Number of slot per pole per phase
rQ Number of rotor slot
sQ Number of stator slot s Slip Λ Airgap Permeance δ Airgap Length N Number of conductors per slot k Time harmonic order v Space harmonic order μ Space harmonic order θ Space angle
0μ Absolute permeability
( )b nI Rotor bar current [A]
brbf Expected frequencies for broken rotor bar [Hz]
,s eccf Expected frequencies for Static eccentricity [Hz]
,d eccf Expected frequencies for dynamic eccentricity [Hz]
,mix eccf Expected frequencies for mix eccentricity [Hz]
stf Expected frequencies for inter-turn short circuit [Hz]
mechf Expected frequencies from mechanical coupling [Hz] ( )p t Instantaneous Power [VA]
ε Eccentric Level a, b, c Phase A, B and C
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APPENDIX
I. Comparison of the airgap length for various types of eccentricities.
Note: Red dot = the motor center, Black cross = the stator center and Blue square= the rotor center
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II. Matlab Script for the Airgap Calculation % Program for calculating the airgap length of eccentricities % clear; % Calculation variable % zeta=0:0.001:2*pi; t=0.15 %parameters% D2= 208.2; % Rotor radius % delta = 1.8; % Symmetrical airgap length % es = -0.15*delta ; %Static eccentricity, [%]*[airgap length], +right, -left er = 0.25*delta ; %Dynamic eccentricity, [%]*[airgap length], +right, -left sp = 0.99*2*pi*50/3; % rotating speed of rotor, (1-s)*2*pi*f1/p % Calculation % n = length(zeta); i=1 for i =1:n ; rs(i) = sqrt((D2+delta)^2-es^2+(es*cos(zeta(i)))^2)+es*cos(zeta(i)); rr(i) = sqrt((D2)^2-er^2+(er*cos(zeta(i)-sp*t))^2)+er*cos(zeta(i)-sp*t); i = i+1; Gap = rs-rr; end A=[Gap;zeta*180/pi]'; [min_gap,I]= min(Gap); min_gap Min_zeta=zeta(I)*180/pi [max_gap,II]= max(Gap); max_gap Max_zeta = zeta(II)*180/pi % polar(zeta,rs,'red') % hold on % polar(zeta,rr) polar(zeta,Gap,'blue')
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III. FEM Simulation
A. Rotor Broken Bars
Number of Position Simulation Case
Broken rotor bars Bar No. Slip
Time [s] A 1 No. 1 1% 2.5 B 5 No.1-5 1% 2.5 C 10 No.1-10 1% 4.5 D 5 No.1-5 0.5% 4.5 E 5 No.1-5 1.5% 4.5 F 5 No. 1,4,7,10,13 1% 2.5 G 5 No. 1,7,13,19,25 1% 2.5 H 5 No. 7,26,28,40,66 1% 2.5
A.1 Varying Operating Slips
• The sideband frequencies are proportional to the twice slip frequency.
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A.2 Varying Numbers of Rotor Broken Bars
• The amplitudes of sidebands depend on the number of broken rotor bars
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A3. Varying Positions of Rotor Broken Bars
• The positions of rotor broken bars have an influence on the amplitude of sidebands
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B. Broken Rotor End Rings
Number of Position Simulation Case
Broken rotor bars Bar No. Slip
Time [s] A 1 No. 1(upper) 1% 2.5 B 4 No.1-4(upper) 1% 2.5 C 10 No.1-10(upper) 1% 4.5 D 1 No.1(upper) 0.5% 4.5 E 1 No.1(upper) 1.5% 4.5 F 2 No. 1(upper) and (down) 1% 2.5 G 2 No. 1(upper) and 2(down) 1% 2.5 H 2 No. 1(upper) and 4(down) 1% 2.5
B.1 Vary Operating Slip
• The sideband frequencies are proportional to the twice slip frequency.
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B.2 Vary Numbers of Broken Rotor End Rings
• The amplitudes of sidebands depend on the number of broken rotor end rings.
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B.3 Vary Positions of Broken Rotor End Rings
• The positions of rotor broken end rings have an influence on the amplitude of sidebands
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C. Static eccentricity
% Static SimulationCase
Eccentricities Slip
Time [s] A 10% 1% 2.5 B 25% 1% 2.5 C 40% 1% 4.5 D 40% 0.5% 4.5 E 40% 1.5% 4.5
C.1 Varying Operating Slip
• The sideband frequencies correspond to the rotor frequency. (The leakage phenomena can be observed due to the unsuitable frequency resolution)
• The rotor slot harmonics also correspond to the operating slip.
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C.2 Varying Static Eccentric Level
• The amplitudes of sidebands and rotor slot harmonics depend on the severity of static eccentricity.
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D. Dynamic eccentricity
% Dynamic SimulationCase
Eccentricities Slip
Time [s] A 10% 1% 2.5 B 25% 1% 2.5 C 40% 1% 4.5 D 40% 0.5% 4.5 E 40% 1.5% 4.5
D.1 Varying Operating Slip
• The rotor slot harmonics considering the dynamic eccentric orders also correspond to the operating slip.
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D.2 Vary Dynamic Eccentric Level
• The amplitudes of rotor slot harmonics considering the dynamic eccentric orders depend on the severity of static eccentricity.
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E. Mix eccentricity
% Static % Dynamic Simulation Case
Eccentricities Eccentricities Slip
Time [s] A 40% 10% 1% 2.5 B 40% 25% 1% 2.5 C 10% 40% 1% 2.5 D 25% 40% 1% 2.5 E 40% 40% 1% 4.5 F 10% 40% 0.5% 4.5 G 10% 40% 1.5% 4.5
E.1 Varying Static Eccentric Level
• The amplitudes of the sidebands as well as the rotor slot harmonics are proportional to the severity of static eccentricity.
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E.2 Varying Dynamic eccentricity Level
• The amplitudes of the sidebands as well as the rotor slot harmonics considering the dynamic eccentric orders are proportional to the severity of dynamic eccentricity.
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E.3 Varying Operating Slip
• The sideband frequencies correspond to the rotor frequency. (The leakage phenomena can be observed due to the unsuitable frequency resolution)
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F. Inter-turn Short Circuit
Simulation Case Number of Short Turn Slip
Time [s]
A 6 turns (3 turns in A+ and 3 turns in A-) 0.5% 4.5
B 6 turns (3 turns in A+ and 3 turns in A-) 1% 4.5
C 6 turns (3 turns in A+ and 3 turns in A-) 1.5% 4.5
D 6 turns (3 turns in A+ ) 1% 2.5
F.1 Varying Operating Slip
• The sideband frequencies correspond to the rotor frequency. (The leakage phenomena can be observed due to the unsuitable frequency resolution)
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F.2 Varying Numbers of Short Turn
• The severity of faults has an influence on the amplitude of sidebands.
G. Combination of Broken Rotor Bars and Broken Rotor End Rings
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IV. Dynamic Model of an Induction Motor Main Diagram [57]
Current Dynamic Box
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Flux Dynamic Box
Torque Calculation Box
Mechanical System Box