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STRAY LOAD LOSSES III INDUCTION MACHINES
Ely
A. A. JIMOH L B. Eng., M. Eng., MIEEE
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A Thesis
£ubmitted to the School of Graduate Studies
in Partial Fulfilment of' the Requirements
for the Degree
Doctor of Philosophy
McMaster University
March 1986
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STRAY LOAD LOSSES IJI INDUCTION MACHINES
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DOCTOR OF PHILOSOPHY (1986).
McMASTER UNIVERSITY
(Electrical & Computer Engineering)
TITLE:
AUTHOR:
, t
Hamilton, Ontario
Stray Load Losses in Induction Machines
-,
Abdul-Ganiyu Adisa Jimoh,
B. Eng. (Ahmadu Bello Uni~ersity)
M. Eng. (Ahmadu Bello University)
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SUPERVISOR:
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Dr. R. D. Findlay
NUMBER.OF PAGES:
xXi,223
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ABSTRACT
Understanaing and minilli'l.zing loss. are the main objectives of·:
seeking solutions to the problems of stray lo.ss in induction machines.
This
thes~s
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contributes towards this
.
problems of stray load loss.
object~ve
by addressing the various
~tions
These problems includ; the
of
definition, origin, components, and effects; theoretical and
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experimental means of evaluation; and loss reduction.
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Insignts intecthese problems are achieved through a
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comprehensive review of the state of the art of the subject.
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We have
established that some
. commonly used terminologies in the subject area
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formed major obstacles to progress in d\>finition.
A
co~ceptually
simple and general theory of squirrel cage
induction machines is presented.
The
th~ory
results in a set of linear
periodic differential equations, which has an infinite number of
~ssible
solutions.
A suitable solution procedure is developed.
Means for accounting for slot
developed;
~penings
and saturation are
These have enabled various field waveforms in a practical
machine to be generated and analyzed.
The analysis produced insights
into the interactions of harmonics and how harmonics contribute to stray
load losses.
Through this, an approximate means of separating a
saturated non-sinusoidal waveform of an air gap flux density waveform
into its fundamental, saturation, and other space harmonics is
developed.
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An expressionrfor determining the machine torque is derived.
This torque expression. the ideas for manipu1ating'and analyzing the
field waveforms. and the presented theory are structured into an
algorithm, modeling the behaviours of
squirre1·~~e
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, __)the algorithm enabled the torque-speed
induction machines.
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~haracteristic
of a practical
machine to be predicted.
The predicted characteristic is compared with that measured by
means of
acce~e~~eter.
The fact that it
the theory and the developed model.
several
advantages,
,.
in~ ud ing
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compares~ery
The new
~teady
well validates
state model has
its easy application to the study and the
.
"evaluation of stray load losses.
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A brief "study of how harmonics
influence the developed torque i~ {lso conducted usi~g ~e)new model.
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Two theoretical methods are developed for predi~ng stray
load loss at the design, manufacturing, or utilization stage of a
machine.
These ~thods. which employ the developed model, are applied
to a practical machine.
The predicted stray losses are compared with the measured, and
that'predicted using the nominal assignation technique.
This draws
attention to the need for experimental investigation of" the SUbject.
Consequently, two experiments, - ca10rimetrlc method and novel experiment
to study inter-bar current problem, are developed.
Due to technical and
economic problems, however. their implementation are not yet completed •.
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ACKlCM.EIlGEIlEIITS
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I would like to express my appreciation to my'supervisor Dr, R.
D. Findlay for his support and-encouragement.
I would like to thank the
other members of my supervisory committee; Drs. R. T. H. Alden and
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LaJ;to.
I have been-very fortunate
i~having
an informal association
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wi th' a distinguished and humble intellectual Professor Michel
Poloujadoff.' His encouragement and many suggestiona have contributed
,
greatly to the
succ~ss
of this work.
Mr. Gcant Neal of the Motor
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Division of Westinghouse (Canada) Limited is gratefully ac~nowledied for
his co-operation and encouragement.
I would also like to thank a very'-
good'friend, Prof. Raymond ~ for his contributions.
T~e
financial supports of Ahmadu Bello University, Zaria,
Nigeria, through an award of a study
fellowshi~;
Canadian
Gov~rnment,
through Canadian Commonwealth Scholarship and Fellowship Program; and
McMaster University are deeply appreciated.
I would. like to thank Dr.
Jennifer Conners, Miss Edna Menzies, Miss'Elsie Lavery, Marion and Perry
Wil~err~~:FOlbigg:
the Philpott
Mem~rial Church,
and the
organizers of the International Student'Fellowship activities for making
,my;family and I feel at home in Canada.
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Their friendship and love will
forever be remembered:
Special thanks should go to Mrs. Maureen McCracken for making
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the beautiful ,drawings, and Mr. Dan McCracken, Dr. Jennifer Conners, and
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Mr. Francis.Omani for proof reading this thesis.
close associ~tes
M'.
The
friendship~f
RaJeev Krishnamurthi. Mr. Franc~s Omani. Dan'a:d
Maureen McCracken.' Mr. Pierre Gauthier. and Mr. Gerry Brown are deeply
appreciated.
I would like to thank
Mrs.~Lynda
Chapple for her diligence
and cheerful co-operation in typing this manuscript.
Last. but
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deftnitely not the least. I would like to thank my children. Bukola·and
Ademola. and my Wife. Gbemisola. for
~heir
patience and support •
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. 2.4- Other Approaches to the Study of Stray
Load Losses
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2.5
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CHAPTER 3
Current Methods (or Reduction of Stray
Load Losses _ "2.5.1 Reduction of Losses in Conductors
2:5.2 Reduction of Losses in the End Region /
2.5.3 Reduction of Harmonics and their
Magnitudes
2,5. 4 Industrial Imperfecti~n
2.5.5 Others
-MODEL DEVELOPMENT
41
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48
50
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3.1
Introduction
3.1.1 Definition of the Problem
50
51
3.2
Stator
54
3.3
Rotor
3,3.1
3.3.2
3.3.3
3.3.4
60
Induced E.M.F. in a Rotor M<!'!Sb> by
the Stator Fiel~s
Induced E.M.F. -in a Rotor Mesh by
the Rotor Fields
Induced C~rrents in the Rotor Winding
An Alternative Formulation
60
63
64
73
3.4
Solution of t-hi' Machine Equation
79
3.5
Effects:Sf Slot Openings
87
3.6
Effects of Saturation
3.6.1 Effects tf Saturation of the Core
3.6.2 Effects of Saturation of the Teeth Body
3.6.3 Effects of Saturation of the Teeth Tips
88
89
89
94
3.7
The Machine Equations
99·
CHAPTER 4
4.1
WAVEFORMS AND HARMONICS:
ON STRAY LOAD LOSS
THEIR INFLUENCE
Magnetic Field of Stator
4.1.1 Stator Fields with Effect of Slot
Openings
4.1.2 Stator Fields with Effects of
·Saturation
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104
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Stator and Rotor Slot Openings and
Saturation
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4.2
CHAPTER 5:
129
Magnetic Fields of the Rotor
4.2.1 The Rotor Impedances
4.2.2 Magnetic Field of the Rotor
131
131
133
4.3' Overall Resultant Air-Gap Field
134
4.4
140
D~scussions.
PREDI~TION
5.1
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5.3
5.4
Harmonics and Stray Load Losses
149
Derivation of .an Expression for Eiectromagnetic
Torque
150
Torque-Time, and Torque-Speed Characteristics
153
5.3.1
Effects of Harmonics
Analysis of the Torque Equation
163-
The Measurement of Torque-Speed Characteristics
5.~.1
Comparison of Predicted and Measured
Torques '
169
157
173
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CHAPTER 6:
6.1
6.2
6.3
6.4
CHAPTER 7:
PREDICTION TECHNIQUES FOR STRAY LOAD LOSS
174
General Concept of Losses in Squirrel-Cage
Induction Machine
175
Method ,. for Predicting Stray Load Loss
,6.2.1 A Criticism of the Conventional Method
of Estimating Total Load Loss
6.2.2 The Proposed Method
179
179
181
Method 2 for Predicting Stray Load Loss
6.3.1 Estimation of Rotor Conductor Loss
6.3.2 The Proposed Method
184
Measurement of Stray Load Loss
6.4.1 Results and Comparison
188
192
CONCLUSIONS
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AND MEASUREMENT OF TORQUE
Suggestions, for Further Works
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REFERENCES
APPENDIX A
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A FAST FOURIER TRANSFORM ALGORITHM·
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('APPENDIX B
CALCULATION OF THE ROTOR BAR IMPEDANCE
PARAMETERS
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B. 1 Resistance
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B.2
Reactances
215
B.3
Skin-Effect
216
A BRIEF DISCUSSION OF THE DEVELOPED
EXPERIMENTS
218
APPENDIX C
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LIST OF PRINCIPAL SYMBOLS
Cross-sectional area
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Flux density
Maximum vth harmonic flux density
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Rotor diameter
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Instantaneous electromotive force (e.m.f.l
supply frequency
Magneto-motive force (m.m.f.l
Magnetic field intensity
Instantaneous current
S~atorls
mth
phas~
instantaneous current
Stator r.m.s. current
p~r
phase
Maximum value of stator current
Magnetizing current
Conductor current
Rotor inertia
Skin effect factor for resistance
Skin effect factor for leakage reactance
Carter's factor or slot permeance factor
Saturation factor
Slot leakage constant
Axial length of machine bore
Length of lamination stack
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Length' of bar
Self inductance
m '
Number of phase
Number of stator phases
M
Mutual inductance
n
Order of harmonic
Synchronous speed in r.p.s.
Number of turns per bar
Number of turns per 'coil
Number of conductor -+
per slpt
p
Number of poles
Pc
No load core loss
Pc1 '
P .
cus' Pscu
Stator copper loss
Pc2' Pcur' . Prcloss' Prcu
Rotor conductor' loss
Pfw' Pf+w
Friction and windage loss
Pi I Pin.
Input power
Pm
:-Iechanical input power or shaft...input power
Output power
Rotational power
Rotor stray load power loss
Rotor iron loss
Stator iron loss
Pscl
Stator core loss
Pseg
Total power loss for the method of segregation
Protorff
Rotor fundamental frequency loss
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Pss
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Stator stray loss due to fundamental
frequency components
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Pstray
Stray load loss
pti
Total power loss
Q
Number of $lots
Q1
Number of stator slots
Q2
Number of rotor slot$
r
Mean air gap radius
Rp
r1
Resistance of a stator winding phase
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s
slip
t
Time
T
Torque (chapt"r 2 onlJ)
T
Time period (1/f)
Ts
Stray load loss torque .(chapter 2 only) _
u
Order of rotor harmonics
v
Instantaneous applied voltage
W
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wlO
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Slot width opening
Stator slot width opening
w20
Rotor slot width opening--
xes
Stator end leakage reactance
x1s
Stator leakage-reactance
xmv
Magnetizing reactance"
Zbv
Vth harmonic bar impedance
Zev
Vth harmonic end-ring segment impedance
"d
Squirrel-cage winding pitch
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Phase 'angle of the vth harmonic component
Electrical conductivity
y
Air gap length
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Angular position
Permeability of free space
Order of stator harmonics
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P
Pole-pitch
, Torque
Flux
Flux linkage
Stator tooth pitch
Rotor tooth pitch
w
Angular velocity
w
Stator angular velocity
w
r
Rotor angular velocity
s
SUBSCRIPTS
at b., c
Stator phases
b
Bar
c
Coil side
e
End-region
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Eddy-current in end-region
em
Electromagnetic
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Full-load
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Air-gap
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Hysteresis and eddy-current losscom~anents
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UST OF FIGURES
FIGURE
to
2.1
. Torque-Speed Characteristic of a
Induction Machine
P~lyphase
3.1
A Simplified conventional equivalent
of an induction machine
3.2
The chain equivalent circuit
3.3
Classification of induction machines parametersc
and variable·s
3.4
Flux. linkages of rotor. mesh with air gap flux due
to a stator sinusoidal component of tho: mm f
3.5
Circular ladder equivalent network for the
squirrel-cage winding
3.6
A network model of the squirr.el cage winding.
3.7(a)
Network of squirrel':cage winding:
the flow of current.
·illustration of
3.7(b)
Network of squirrel-cage winding:
current in the bars
calculation of
3.8
Calculation of fictitious opening of a semi-closed
slot
3.9
Calculation of fictitious opening
4.1
Layout· of the stator's full-pitch concen:ric winding
4.2
Step. distribution of the stator's air-gap flux
density distribution
~.3
Spectrum of three phase field waveform of figure 4.2(a)
109
4.4
Stator punching
1 12
4.5
The rotor punching
circ~it
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a closed slot
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4.6
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Stator current vs. effect1ye slot openings
4.7
Slots permeance factor distributions ,and spectral
115
4.8
Stator air gap flux density with s10t ripples
117
4.9
B-H'mag~:tization
122
4.10
Saturation factor versus MMF
4.11 .
Stator air-gap' flux density waveforms with saturation,
and their spectra! '.
124 .
A study of how saturation Influences the air gap
field waveforms
126
Stator air-gap flux density w~veforms with saturation
and stator and rotor slot ripples
130
4.14
A. typical shape of a double cage rotor slot
133a
4.15
An equivaient circuit for the double cage of figure
4.14
133a
4.12
4.13
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114
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4.16
curves
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The waveform, and its spectrum, of the rotor air gap
flux density with saturation and stator and rotor slot
ripples
.
4.17
Saturation factor distrtbution
138
4.18
The stator, rotor, and the overall resultant air gap
flux density waveforms with their 'spectral
139
of harmonics in. induction motors
135
141
4.19
CI~ssification
5.1
Tor-que versus time
156
5.2
Torque-speed characteristics
158
5.3
Torque vs. time
161
5.4
Block diagram of accelerometer connections
6.1
Generalized
real
power flow in indu.ction motors
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6.2
Real power flow at no-load
185
6.3
Actual real- power flow at full-load
186
6.4
Apparent real power flow 'at full-load
186
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6.5
Set-up forwno-load test
6.6..
Se~-up
6.7
Set~up
C. 1
machine "iaminations
method exper~me.n~""
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for D.C. test
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I'nstrume~ti'ri'g
for
full~load
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Instruinenting a 'rot'or lamination' for a st~dy 'of the
current problem
C.2,
t~ter-bar
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LIST OF j'ABLES
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'TABLES
2. 1
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2.2
,2.3
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3.1
PAGE
Components of stray load l,?sses as given by
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Schwarz [4]
12
Classification of me,thods for measuring stray load
losses
17
Comparison of measurement approaches
19
List of Parameters and variables
phenomenon of induction machines.
Typical
mach'ine
4.2
det~ils
parti~ipating
in the
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of a practical.squirrel-cage induction
'05
Significant harmonics of stator air gap waveforms
(Amplitude") 0.1 Tesla)
11~
4.3
Effects of non-smooth. air-gap on stator air-gap fields
119
4.4
Magnitudes of some harmonics of stator slot ripple
120
4.5'
Effects of saturation on stator air gap field
128
4.6
Stator fields and effects of both stator and rotor
slot ripples and saturation .
132
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4.7
Significant harmonics of rotor
(Amplitude) 0.1 Tesla)
a~
gap waveforms
136
4.8(a)
Full-load case (I 1 = 26.7A. s=O.O)
143
4.8(b)
Locked-rotor case at-locked-rotor current
(I, = i33.97A. s=,.O)
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144
Locked-rotor case· at full-load current
(I 1 = 26.7A. s=1.0)
145
4.8(c)
4.8(d)~-10ad
5. 1
case (I, = 7.21A. s=0.0005)
Effects of
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159
harmo~ics
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