University of WollongongResearch Online
University of Wollongong Thesis Collection University of Wollongong Thesis Collections
1987
Some aspects of analysis and design of single-framecascaded induction machinesB. S.P. PereraUniversity of Wollongong
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Recommended CitationPerera, B. S.P., Some aspects of analysis and design of single-frame cascaded induction machines, Doctor of Philosophy thesis,Department of Electrical and Computer Engineering, University of Wollongong, 1987. http://ro.uow.edu.au/theses/1350
SOME ASPECTS OF ANALYSIS AND DESIGN
OF SINGLE-FRAME CASCADED INDUCTION MACHINES
A thesis submitted in fulfilment of the
requirements for the award of the degree of
DOCTOR OF PHILOSOPHY
from
THE UNIVERSITY OF WOLLONGONG
by
B.S.P. PERERA, BSc (Eng) S. Lanka, MEngSc N.S.W.
Department of Electrical and
Computer Engineering, 1987.
Dedicated to
my parents, my wife and children
TABLE OF CONTENTS
i
ACKNOWLEDGEMENTS vi
ABSTRACT vii
LIST OF PRINCIPAL SYMBOLS ix
CHAPTER 1: INTRODUCTION 1
1.1 The Single-Frame Cascaded Induction Machine (SFCIM) 1
1.2 A Survey of Related Research 4
1.3 Identification of Work Presented in this Thesis 13
CHAPTER 2: STEADY-STATE CIRCUIT THEORY OF SFCIM 15
2.1 Introduction 15
2.2 Assumptions 18
2.3 Formation of the Coupling Impedance Matrix for SFCIM
having a Multicircuit Single-Layer Bar Rotor Winding 20
2.4 Evaluation of Torque 42
2.4.1 Asynchronous mode of operation 42
2.4.2 Synchronous mode of operation 48
2.5 Formation of the Coupling Impedance Matrix for SFCIM
having a Single-Layer Phase-Wound Rotor Winding 50
2.6 Application of the Circuit Theory of SFCIM 52
2.6.1 Introduction 52
2.6.2 Description of the experimental SFCIM 52
2.6.3 SFCIM having a multicircuit single-layer
bar rotor winding 55
2.6.3.1 The coupling impedances 55
2.6.3.2 Current/torque versus slip curves 57
11
2.6.4 SFCIM having a single-layer phase-wound rotor
winding 64
2.6.4.1 The coupling impedances 64
2.6.4.2 Current/torque versus slip curves 65
2.7 Summary and Conclusions 69
CHAPTER 3: EXPERIMENTAL DETERMINATION OF PARAMETERS OF
THE EQUIVALENT CIRCUIT OF AN SFCIM HAVING
A PHASE-WOUND ROTOR 71
3.1 Introduction 71
3.2 Experimental Determination of Equivalent Circuit
Parameters 74
3.2.1 Introduction 74
3.2.2 Determination of shunt parameters 75
3.2.3 Determination of series parameters 79
3.2.4 Experimental results 83
3.2.4.1 Shunt parameters 83
3.2.4.2 Series parameters 90
3.2.4.3 Experimental and theoretical
equivalent circuits 93
3.2.4.4 Performance characteristics 98
3.3 Effects of the Redistribution of Leakage Reactance
on Performance 99
3.4 Summary and Conclusions 103
CHAPTER 4: SOME ASPECTS OF IRON LOSSES IN SFCIM 105
4.1 Introduction 105
4.2 Frequencies of Flux Pulsations in Stator and Rotor 106
4.3 Flux Density Waves in the Airgap and Yoke 107
iii
4.4 Eddy Current Losses in Stator and Rotor 108
4.4.1 The analytical expressions 108
4.4.2 Theoretical results 115
4.5 Hysteresis Losses in Stator and Rotor 118
4.5.1 The analytical expressions 118
4.5.2 Theoretical results 124
4.6 Experimental Determination of Iron Losses 125
4.6.1 Introduction 125
4.6.2 Iron losses under standstill conditions 127
4.6.3 Iron losses under running conditions 130
4.7 Summary and Conclusions 136
CHAPTER 5: DESIGN ASPECTS OF PHASE-WOUND ROTOR WINDINGS
FOR SFCIM 138
5.1 Introduction 138
5.2 Single- and/or double-Layer Smoothly/Discretely
Distributed Phase-Wound Rotor Windings 143
5.3 Maximum Possible Torque Output from a Given Frame
Size from Magnetic Field Viewpoint 146
5.3.1 Dependence on winding factors 146
5.3.2 Dependence on pitch and spread 153
5.3.3 Conclusions 157
5.4 Magnetising Volt-Ampere Requirements 159
5.5 Maximum Possible Torque per Magnetising Volt-Ampere 162
5.5.1 Dependence on winding factors 162
5.5.2 Dependence on pitch and spread 164
5.5.3 Conclusions 167
iv
5.6 Harmonic Leakage Reactance of Phase-Wound Rotor
Windings 167
5.6.1 Dependence on pitch and spread 167
5.6.2 Conclusions 170
5.7 Design of p- and q-Pole Pair Stator Windings and
Evaluation of Performance 174
5.7.1 Design of stator windings 174
5.7.2 Evaluation of performance 177
5.8 Conclusions 180
CHAPTER 6: THE SFCIM AS A BRUSHLESS CONSTANT-FREQUENCY
VARIABLE-SPEED GENERATOR 182
6.1 Introduction 182
6.2 Variation of Slip with Angular Speed 183
6.3 Real and Reactive Power Flows 186
6.4 Generating Scheme for Windmill Applications 187
6.5 Theoretical Analysis 189
6.6 Theoretical and Experimental Results 191
6.7 Summary and Conclusions 202
CHAPTER 7: SUMMARY AND CONCLUSIONS 204
7.1 Conclusions 204
7.2 Suggestions for Future Research 207
BIBLIOGRAPHY 209
APPENDIX 1: ROTOR CIRCUIT FLUX LINKAGES 214
1.1 Total Flux Linkage with a Cage Mesh due to Rotor
Currents 214
V
1.2 Total Flux Linkage with a U-loop due to Rotor
Currents 218
APPENDIX 2: EXPRESSIONS FOR TORQUE 223
APPENDIX 3: THE WOUND ROTOR INDUCTION MACHINE AS A
CONSTANT-FREQUENCY VARIABLE-SPEED GENERATOR 227
APPENDIX 4: HARMONIC CONTENT OF MMF WAVEFORMS PRODUCED BY
MULTICIRCUIT SINGLE-LAYER BAR ROTOR WINDING 231
vi
ACKNOWLEDGEMENTS
The author wishes to express the deepest appreciation to his
Supervisor, Professor B.H. Smith of the Department of Electrical and
Computer Engineering, The University of Wollongong, for the
guidance, support and encouragement given throughout this research.
The inspiration given by Professor N.S. Karunaratne of the
Department of Electrical Engineering, University of Moratuwa,
Sri Lanka, to undertake postgraduate studies is greatly appreciated.
Special thanks are due to Dr J.R. Lucas for the encouragement given
to continue the research work. The encouragement given by
Dr M.P. Dias is also greatly appreciated.
The author wishes to express his appreciation to Mr J.F. Willis
of the Department of Electrical and Computer Engineering, The
University of Wollongong, for assisting with the construction of the
experimental machines and other auxiliary devices. Thanks are also
due to all the Technical Staff in the Department, for the generous
assistance given at various stages of this research. The assistance
given by Miss M.J. Fryer in reading the manuscript is warmly
appreciated. Her abilities in this regard are unsurpassed.
Finally, the author acknowledges with deepest gratitude the
patience, understanding and continuous support rendered by his wife,
Swarna, and children, Lakshal and Ruvandhi.
vii
ABSTRACT
A Single-Frame Cascaded Induction Machine (SFCIM) is a
brushless single-unit version of the well known cascade connected
two-machine system. Applications of the SFCIM include, slip-power
recovery for speed control; operation as a high power stepping
motor; and generation of constant frequency in both aircraft and
wind energy applications.
A new steady-state circuit theory for the SFCIM having a
'multicircuit single-layer bar rotor winding' is described.
Assumptions made in the development of the new model are no
different to those made in the analysis previously existing and yet
the new theoretical model is relatively simple in comparison. It is
based on the analysis of coupled electric circuits having
time-varying coefficients and the complete machine is represented by
a coupling impedance matrix having elements which are simple to
calculate. The experimental results are compared with the
theoretical results calculated using the new model. It is shown
that this model also can be applied to SFCIMs having phase-wound
rotors,
It is hypothesised that an SFCIM having a phase-wound rotor
winding can be represented by the series-connected equivalent
circuits of two conventional induction machines. The no-load and
locked-rotor tests performed on conventional induction machines
cannot be applied in this case because of the extra loop present in
the equivalent circuit. Techniques suitable for the experimental
determination of its parameters are described and the experimentally
determined equivalent circuit is compared with the theoretically
calculated equivalent circuit. Performance characteristics
predicted using these two circuits are compared with the
experimentally observed characteristics.
viii
Some theoretical aspects of iron losses in an SFCIM are
considered which are supplemented by experimental results. The
component of iron losses considered in the theoretical work is only
due to the two fundamental flux density waves. In the asynchronous
operation of the SFCIM as a motor, it is shown that the supply of
the large pole pair stator side at constant frequency, while
allowing the small pole pair stator side to carry the slip frequency
currents, leads to lower iron losses.
The wide ratio pole combination of 1 and 3 gives the highest
possible cascade synchronous speed for an SFCIM. In this case the
'multicircuit single-layer bar rotor winding' is unsatisfactory.
The phase-wound rotor windings are better suited as their design is
more flexible. Some design aspects of these windings are
considered, giving emphasis to optimisation of certain performance
characteristics. It is shown that the minimisation of harmonic
leakage reactance of the phase-wound rotor winding plays an
important role in the design.
As a brushless generator the SFCIM is suitable for
constant-frequency variable-speed generation in aircraft and wind
energy applications. Steady-state, experimental and theoretical
characteristics of operation, of the SFCIM, as a variable-speed
constant-frequency generator are presented. The theoretical
analysis of this mode of operation is carried out using the new
steady-state circuit theory presented.
IX
LIST OF PRINCIPAL SYMBOLS
breadth of a phase band, slots
2p-pole peak airgap flux density, T
2q-pole peak airgap flux density, T
coil pitch, slots
airgap diameter, m
2p-pole stator supply frequency, Hz
2q-pole stator supply (induced) frequency, Hz
frequency of the rotor induced voltage, Hz
airgap length, m
instantaneous current, A
instantaneous current in the cage mesh of k phase
group, A
instantaneous current in the h U-loop of k phase
group, A
peak (rms) value of the current in 2p-pole stator, A
phasor representing the current in phase 'a' of
2p-pole stator, A
peak (rms) value of the current in 2q-pole stator, A
phasor representing the current in phase 'a' of
2q-pole stator, A
peak (rms) value of the current in a cage mesh of
any phase group of the rotor, A
per phase current (rms) of the phase-wound rotor, A
phasor representing the current in the cage mesh of
k phase group of the rotor, A
phasor representing the current in a phase winding
of the phase-wound rotor, A
phasor representing the phase current of the rotor
referred to 2p-pole stator, A
peak (rms) value of the current in an h U-loop of
any phase group of the rotor, A
phasor representing the current in the h U-loop of
k phase group of the rotor, A
running integer
rotor phase group reference number
rotor leakage factor
2n-pole winding factor of a phase winding of the
rotor
slot-leakage inductance of a bar, H
total slot-leakage inductance of a phase winding of
the rotor, H
cyclic inductance of a cage mesh, H
end-ring inductance of a cage mesh, H
total overhang leakage inductance of a phase winding
of the rotor, H
overhang inductance of an m U-loop of any phase
group, H
4- V»
cyclic inductance of an h U-loop, H
core length, m
airgap inductance of a cage mesh, H
XI
airgap inductance of a phase winding of the rotor, H
self-inductance of a phase winding of the 2p-pole
stator, H
self-inductance of a phase winding of the 2q-pole
stator, H
self airgap inductance of an m U-loop of any phase
group, H
mutual airgap inductance between any two phase
windings of the rotor, H
mutual airgap inductance between any two cage
meshes, H
mutual airgap inductance between the cage mesh of
k phase group and the m U-loop of the n phase
group, H
end-ring inductance common to a cage mesh of a phase
group and the h U-loop of the same phase group, H
mutual overhang inductance between the m U-loop
and the h U-loop of the same phase group, H
maximum value of the fundamental mutual inductance
between a phase winding of the 2p-pole stator and a
cage mesh of any phase group of the rotor, H
maximum value of the fundamental mutual inductance
between a phase winding of the 2p-pole stator and a
phase winding of the rotor, H
maximum value of the fundamental mutual inductance
between a phase winding of the 2p-pole stator and an x. U
m U-loop of any phase group of the rotor, H
Xll
M - maximum value of the fundamental mutual inductance qr between a phase winding of the 2q-pole stator and a
cage mesh of any phase group of the rotor, H
M - maximum value of the fundamental mutual inductance qr
between a phase winding of the 2q-pole stator and a
phase winding of the rotor, H
M - maximum value of the fundamental mutual inductance qrm
between a phase winding of the 2q-pole stator and an
+• Vi
m U-loop of any phase group of the rotor, H
M , M - mutual inductance between any two phase windings of sp P
the 2p-pole stator, H
M , M - mutual inductance between any two phase windings of sq q
the 2q-pole stator, H
M , , - mutual airgap inductance between m U-loop of the ukm,unh
k phase group and the h U-loop of the n phase
group, H
M - mutual airgap inductance between m U-loops of n ukm,unm
and k phase groups, H
n - pole pair number of rotor space harmonics
p - pole pair number
q - pole pair number
r, - resistance of a bar in the cage mesh, ft b
r, - total series resistance of a phase winding of the b
rotor, ft
r - resistance of the end-ring section of a cage mesh, ft ce
Xlll
- resistance of the end-ring section common to a cage
mesh of a phase group and the h U-loop of the same
phase group, ft
) - 2p-pole (2q-pole) component rotor winding resistance
referred to the 2p-pole stator, ft
) - 2p-pole (2q-pole) stator winding resistance referred
to the 2p-pole stator side, ft
- resistance of the h U-loop excluding the end-ring
section, ft
) - 2p-pole (2q-pole) component core loss resistance
referred to the 2p-pole stator, ft
- resistance of a phase winding of the 2p-pole stator,
ft
- resistance of a phase winding of the 2q-pole stator,
ft
- composite slip (s = s s )
- slip with respect to the 2p-pole stator field
- slip with respect to the 2q-pole rotor field
- time, s
- torque, Nm
- total 2p-pole component of torque, Nm
- total 2q-pole component of torque, Nm
- peak (rms) value of the applied voltage on the
2p-pole stator, V
- phasor representing the applied voltage on phase la'
of the 2p-pole stator, V
XIV
- peak (rms) value of the applied voltage on the
2q-pole stator, V
- voltage V referred to the 2p-pole stator, V
- phasor representing the applied voltage on phase 'a'
of the 2q-pole stator, V
- phasor representing the 2q-pole stator applied
voltage referred to the 2p-pole stator, V
) - 2p-pole (2q-pole) component rotor leakage reactance
referred to the 2p-pole stator, ft
) - 2p-pole (2q-pole) stator leakage reactance referred
to the 2p-pole stator, ft
- 2p-pole fundamental magnetising reactance referred
to the 2p-pole stator, ft
) - 2q-pole fundamental magnetising reactance referred
to the 2q-pole (2p-pole) stator, ft
- number of U-loops within any phase group of the
rotor
- breadth of a phase band, rad, deg.
- half the average coil pitch, rad, deg.
- mechanical angle between the axis of the k phase
group of the rotor and a reference axis of phase 'a'
of 2p-pole stator, rad
_7 - permeability of free space, 4.77x10 H/m
2 - total airgap surface area, m
- flux linkage, Wb
XV
TP - total fundamental airgap flux linkage with phase 'a'
of the 2p-pole stator due to all rotor mesh
currents, Wb
TP - total fundamental airgap flux linkage with phase 'a' aq
of the 2q-pole stator due to all rotor mesh
currents. Wb
r th V . - total flux linkage with the cage mesh of k phase ck
ukm
group due to all rotor mesh currents, Wb
TP r - fundamental airgap flux linkage with the cage mesh
of k phase group of the rotor due to currents in
the 2p-pole stator, Wb
«psPa - fundamental airgap flux linkage with the cage mesh ck
of k phase group of the rotor due to current in
phase 'a' of the 2p-pole stator, Wb
TPS^ - fundamental airgap flux linkage with the cage mesh CK
of k phase group of the rotor due to currents in
the 2q-pole stator, Wb
5psqa - fundamental airgap flux linkage with the cage mesh ck
of k phase group of the rotor due to current in
phase 'a' of the 2q-pole stator, Wb
Tpr, - total flux linkage with the m U-loop of k phase ukm
group due to all rotor mesh currents, Wb
Tpsf - fundamental airgap flux linkage with the m U-loop
of k phase group due to currents in the 2p-pole
stator, Wb
XVI
fundamental airgap flux linkage with the m U-loop A. u
of k phase group due to currents in the 2q-pole
stator, Wb
angular velocity of the rotor, rad/s
angular supply frequency of the 2p-pole stator,
rad/s
angular supply (induced) frequency of the 2q-pole
stator, rad/s
angular frequency of the rotor induced voltage,
rad/s
1
CHAPTER 1: INTRODUCTION
1.1 The Single-Frame Cascaded Induction Machine (SFCIM)
The well known cascade connection of two wound rotor induction
machines is shown in Figure 1.1. The shafts of the machines are
coupled mechanically and the rotor windings are connected together
electrically.
2p-pole
machine
supply
tt
X
2q-pole
no. chine
short circuit
FIGURE 1.1 CASCADE CONNECTION OF TWO WOUND ROTOR INDUCTION
MACHINES
Depending on the sequence adopted in the interconnection of the
rotor windings, the two machine system will be equivalent to a single
induction machine with a pole pair number of either (p+q) or (p-q).
The latter pole pair number corresponds to the 'differential' cascade
connection in which the torque developed by individual machines
oppose each other and hence has no technical interest attached to it.
2
In the former connection, it can be discovered that the magnetic
fields produced by the rotor windings of the two machines are
contra-rotating with respect to the common shaft and both machines
contribute to useful power. If the 2p-pole stator supply angular
frequency is w rad/s (w = 2tff ), and if the angular velocity of the P P P
rotor is w rad/s in the same direction as the synchronously rotating m
stator field of the 2p-pole machine, the slip (s ) of the 2p-pole Mr
machine is
= 1 - p -Ji (1.1) p w
The angular frequency of the rotor induced voltage (w ) is given by
w = s w (1.2) r P P
With respect to the shaft, the angular velocity of the 2q-pole
field produced by the rotor is -s w /q and with respect to the stator P P
it is (w -s w /q). Therefore, with respect to the 2q-pole field m p p
produced by the rotor, the slip (s ) of the 2q-pole machine is given
by
w_ q s q s w H P P
m + 1 (1.3)
3
The angular frequency (w ) of the 2q-pole stator induced voltage is
given by
w = - s s w (1.4) q q p p v '*'
Multiplying Equation (1.3) by s to and replacing s of the right hand
side of the result by Equation (1.1) yields
w w = —J— (1 - s s ) (1.5) m p+q p q
Thus the synchronous speed of the cascade connection (i.e. when
s s = 0) is to /(p+q). p q p Substituting w of Equation (1.5) in Equation (1.1) and
replacing the product s s by s (the composite slip)
s = _i_L_£P (1<6) p p + q
The particular mode of operation depicted in Figure 1.1 is
referred to as an 'asynchronous' mode of operation. If, instead of
the short-circuit, the 2q-pole stator is also supplied at an angular
frequency of w together with the 2p-pole stator supply at the
angular frequency of w , the speed of operation of the machine is
given by
w + to u, = P , q (1.7)
m p + q
which is referred to as 'doubly-fed' mode of operation. It can also
4
be classified as a 'synchronous' mode of operation, as the rotor
speed is entirely governed by w and w . For the specific case in P q
which w = 0, the 2q-pole stator can be supplied with direct current T.
and the machine operates synchronously.
The single-frame cascaded induction machine (SFCIM) is a
single-unit version of the two machine cascade connection; making it
a brushless machine. Its stator and rotor windings share a common
magnetic structure. Nevertheless, the fundamental relationships
given by Equations (1.1) through to (1.7) are equally applicable to
the SFCIM.
1.2 A Survey of Related Research
Experiments on a cascade connection of induction machines for
1 2 low-speed operation were first carried out by Steinmetz and Gorges .
It is understood from Reference 5 that Lydall was the first to
suggest that a single-unit machine could be built by selecting
dissimilar pole pair numbers for each machine in the cascade
connection. Such a machine would have two stator windings and two
rotor windings and its performance would be impaired by the effects
of high leakage fluxes and winding resistances. A most remarkable
3 4 achievement was that of Hunt ' who demonstrated that the SFCIM could
be built with a single stator winding and a single rotor winding.
The single stator winding developed by Hunt had two parallel paths.
Its connections were such that slip-frequency currents could flow
only when paths outside the winding were provided, whereas the
supply-frequency currents could flow in the winding independent of
the outside paths. He showed that the two pole pair numbers selected
5
must differ by more than 2 to eliminate the unbalanced magnetic pull
on the rotor. Hunt devised his single rotor winding by selective
elimination of conductors of two conventional windings of appropriate
pole numbers placed in the same slots. This winding was superior in
5 performance. Later, Creedy also contributed towards development of
the SFCIM by devising new stator and rotor windings.
The single-unit, pole-amplitude modulated (P.A.M.) induction
frequency converter developed by Broadway and Thomas had a close
resemblance to Hunt's machine. This frequency converter consisted of
two independent stator windings of different pole pair numbers. The
rotor winding was adapted from the P.A.M. stator windings designed
for two speed operation of induction motors. It had two parallel
paths per phase for one pole pair number and served as a
short-circuited winding for the other pole pair number. Another form
of a single rotor winding specifically designed for a pole
combination ratio of odd/even, or even/odd, is described in
Reference 6.
The work on the single-unit frequency converters motivated
Broadway and Burbridge to examine the new improvements to Hunt's
machine. A comprehensive discussion on the new forms of stator and
rotor windings suitable for SFCIMs is given in Reference 7. A new
addition to the existing types of stator windings is the
'Multi-Parallel-Path' winding, which can be designed for any pole
combination except when one pole pair number is an integral multiple
of the other. P.A.M. stator windings on the other hand can be
designed for any pole combination.
6
It is significant that the new types of rotor windings proposed
in Reference 7 are all based on the harmonic behaviour of a squirrel
cage rotor having (p+q) bars. When such a rotor is excited by a
rotating flux density wave of 2p-poles, the pole pair number of the
resulting rotor slot harmonics is given by the well known
relationship
(1.8)
where n = pole pair number of the rotor slot harmonic
k = running integer.
It can be seen that for k = 0, the rotor produces a positive-sequence
2p-pole wave in response to the exciting flux density wave. For
k = -1, the rotor produces the required negative-sequence 2q-pole
wave. In addition to these desired waves, a series of unwanted space
harmonics are produced. An SFCIM of this form was first described by
o
Broadway, Fong and Rawcliffe . Naturally, the magnitudes of the
unwanted rotor slot harmonics can be reduced or eliminated by
adopting a distributed winding having (p+q) phases. Reference 7
describes two new types of (p+q)-phase distributed rotor windings
which are illustrated in Figure 1.2.
n = P 1 + (P+q)
1.2.1 (p+q)-PHASE SHORT CIRCUITED DOUBLE-LAYER WINDING
1.2.2 (p+q)-PHASE MULTICIRCUIT SINGLE-LAYER BAR WINDING
FIGURE 1.2 (p+q)-PHASE ROTOR WINDINGS FOR SFCIM
Figure 1.2.1 shows a double-layer winding and its design is
based on selective elimination of coils from each phase group in
order to increase the effectiveness of the total winding with respect
to the two pole numbers of interest. In the process, some slots
would be left unoccupied, leading to under-utilisation of the
8
available slot space. The synthesis of the winding shown in
Figure 1.2.2, referred to as the 'Multicircuit single-layer bar
winding', is fully described in Reference 7. This winding is
suitable for close ratio pole combinations. Being a single-layer
winding it has the advantages of simplicity and robustness in
construction. Much of the analysis given in Reference 7 refers to
the steady-state operation of the SFCIM with this single-layer
winding.
In Reference 7, Broadway and Burbridge have shown a particular
interest in the (2+6)-pole SFCIM having the above single-layer
winding, as this pole combination gives the highest possible cascade
synchronous speed for a given supply frequency of t» . Due to the XT
wide ratio pole combination of 3:1 the effectiveness of this rotor
winding with respect to both pole combinations is not of the same
order, and hence, a satisfactory performance cannot be expected from
the machine.
Instead of utilising the electrical asymmetry of the (p+q)-phase
rotor winding, it is also possible to use the magnetic asymmetry of
the rotor structure itself in order to produce the necessary
positive-sequence p-pole pair wave and the negative-sequence q-pole
pair wave. An induction machine utilising this property has been
q
developed by Broadway . This cageless induction machine has a rotor
structure which has 'directional' magnetic properties. As a low
speed asynchronous motor the machine is self-starting but the
performance of this machine in this mode of operation is seriously
impaired by saturation and space harmonic effects. Synchronous
9
operation by doubly feeding has been found to give much superior
performance.
In the rotor windings described earlier, the electrical or the
magnetic asymmetry of the (p+q)-phase rotor windings with respect to
a 2p-pole exciting wave has been exploited in order to produce the
required 2q-pole wave. This effect is present in a much simpler case
where one of the rotor phases of a wound rotor induction motor is
unbalanced when compared to the others. With a heavy imbalance the
motor would run at near half synchronous speed, a phenomena which was
first observed by Gorges . This behaviour is due to the
negative-sequence field produced by the asymmetrical rotor having a
pole pair number equal to the exciting stator field. The
low-frequency currents flowing in the mains supply make this
half-speed machine unacceptable in practice. At exactly half
synchronous speed the negative-sequence field produced by the rotor
is stationary with respect to the stator. Schenfer demonstrated
that by having a second stator winding which carries direct current,
the machine can be synchronised to run at half synchronous speed.
12 Russel and Norsworthy enhanced Schenfer's scheme of operation by
designing stator windings which carry both alternating and direct
current. An improved version of this mode of operation was described
13 by Broadway and Tan . Their machine, referred to as the 'Brushless
stator-controlled synchronous-induction machine', circumvents the
disadvantages associated with both synchronous and asynchronous modes
of operation of earlier versions. The rotor windings utilised by the
machine are identical to the rotor windings described in Reference 7,
however they are expected to produce positive- and negative-sequence
10
waves of the same pole number. Nevertheless, the configuration and
the mode of operation of the machine has a close resemblance to the
SFCIM.
The SFCIM described in Reference 7 is also capable of
functioning as an efficient high-frequency alternator. Broadway,
14 15 Cook and Neal ' have considered the operation of a (10+6)-pole
SFCIM as an alternator operating under both separate- and
self-excitation. Several voltage regulating schemes have been
considered.
Another area of investigation has been the 'doubly-fed'
operation of cascade connected induction motors, the principle of
which is outlined in Section 1.1. The steady-state characteristics
IK 17
of such a scheme has been investigated by Smith ' . The
instabilities existing in potentially useful operating regions have
18 19 been examined by Cook and Smith ' . This study on the dynamic
characteristics has demonstrated stable operation of the SFCIM
incorporating several different feedback schemes. A useful
application of the doubly-fed cascade system is as a high power
18 19 20 21 stepping motor ' ' ' . It is clear from Equation (1.7) that if
w = -w , the rotor remains stationary and if the phase of one supply P Q is altered with respect to the other, the torque would be developed
at standstill due to the change in the internal torque angle of the
machine. Continuous change of phase corresponds to a change of
frequency and hence it would be possible to smoothly make the change
from stepping mode to continuous speed of rotation. The variable
phase supply is a d.c. link inverter and its drive logic is such that
forward or reverse motion is possible. The step size is a function
11
only of the initial logic design and hence is fundamentally different
from usual stepping motors. The limitations which are applicable to
reluctance type stepping motors as their sizes increase are not
applicable in the present case.
Speed control of wound rotor induction motors by slip-power
control similarly has been extended to SFCIMs. The prime advantage
in this case being the brushless nature of such systems.
Availability of static power converters has led to the
revitalisation of the old Scherbius and Kramer systems.
Sub-synchronous operation is possible by recovery of slip-power via
a static rectifier and an inverter. Reference 22 deals with such a
scheme developed for the control of 1650kW induction motor pump
drives. This scheme suffers from inability to regenerate at
sub-synchronous speeds. The 'self-controlled reversible frequency
23 converter' scheme, developed by Nonaka and Oguchi , has overcome
this difficulty and the operation is only upper-bound in speed.
Hence super-synchronous operation is possible.
24 Kusko and Somuah developed a similar system of speed control
using an SFCIM having two separate windings on both rotor and stator.
25 More recently, Oguchi and Suzuki developed a static Kramer system
of speed control based on an SFCIM having a 2:1 pole changing
Dahlander winding as the stator winding and a (p+q)-phase,
phase-wound rotor winding. This scheme, with the absence of a line
commutated inverter, requires less reactive power and injects less
harmonic currents into the supply system. Experimental results of
the performance of this scheme are given in Reference 25, but no
12
theoretical modelling has been undertaken due to difficulties in
estimating the parameters.
Another form of a brushless generator has been investigated
9 fi 97 9ft
recently by Ortmeyer and Borger ' ' . The principle of operation 29 of this form of generator stems from the work of Riaz . In
aero-space applications, the shaft speed available for power
generation is variable and hydraulics are generally employed to
9K 97 convert the variable speed to a constant speed ' . The system
reliability has been less than what is desired, and a definite
improvement is possible if the constant frequency is directly
9K 97 obtained from the variable shaft speed. Ortmeyer and Borger '
employed cascade connected induction machines directly connected to
the variable speed shaft with one of the stator windings excited by a
static inverter. As the shaft speed changes the inverter frequency
has to be adjusted in order to maintain the second stator at constant
frequency. The fundamental frequency relationships of this mode of
operation can be easily derived from Equations (1.1) through
to (1.7). By operating the system over a limited speed range around
the cascade synchronous speed, the real excitation power which has to
be supplied by the inverter would be only a fraction of the
electrical power delivered to the load. Reference 26 examines some
basic considerations of a doubly-fed two machine system, and
Reference 27 investigates the closed-loop control of the total
generating system. No attempt, however, has been taken to present
the steady-state characteristics of the scheme operating over a wide
speed range. The same scheme can be equally applied in windmill
applications where the shaft speed is a variable.
13
1.3 Identification of Work Presented in this Thesis
This thesis presents material in relation to some developments
of the SFCIM. Areas considered include analysis, design and
applications. A new steady-state circuit theory for the SFCIM having
7 a 'multicircuit single-layer bar rotor winding' is presented in
Chapter 2. In the development of this model the concept of 'cyclic'
30 inductance, which has been extensively used by Poloujadoff is
utilised. The model can be conveniently applied to analyse any
balanced (p+q)-phase rotor winding configuration for which the
concept of 'cyclic' inductances can be applied. Application of this
model is extremely simple compared to that developed by Broadway and
7 Burbridge . Theoretical results obtained using this new model are
presented together with experimental results.
No technique has so far been developed for the experimental
determination of the parameters of the equivalent circuit proposed
for SFCIMs having phase-wound rotors. Chapter 3 presents the
theoretical analysis behind the experimental procedures which can be
applied to identify the equivalent circuit parameters. The
performance predicted using the experimentally determined equivalent
circuit parameters is compared with actual and theoretical
performance.
The problem of iron losses in SFCIMs is more complex than that
of a conventional induction machine, as a result of the co-existence
of the 2p- and the 2q-pole fields. The modelling of iron losses
carried out in Chapter 4 is by no means rigorous, and it is only
intended to give an insight into the distribution of the iron losses
in the teeth and the yoke of both stator and rotor under running
14
conditions. This study is supplementary to the work in Chapter 3
which was concerned with the experimental determination of parameters
of the equivalent circuit.
The multicircuit single-layer bar winding is not very suitable
for SFCIMs having wide ratio pole combinations such as in the case
of the (2+6)-pole machine. As an alternative, the design of
phase-wound rotor windings is considered in greater detail in
Chapter 5. Theoretical investigations presented in this chapter are
specifically related to the (2+6)-pole SFCIM.
The steady-state characteristics of the SFCIM as a
variable-speed constant-frequency generator are examined in
Chapter 6. It is shown that the new steady-state circuit theory
developed in Chapter 2 can be conveniently applied to analyse the
variable-speed constant-frequency situation. Theoretical results are
compared with experimental results.
15
CHAPTER 2: STEADY-STATE CIRCUIT THEORY OF SFCIM
2.1 Introduction
A steady-state circuit model for a cascade connected set of
induction machines simulating an SFCIM can be conveniently obtained
by a series connection of the individual equivalent circuit of each
machine. This simple arrangement has been extensively used in the
past. In single-unit cascaded induction machines, when each phase
of the (p+q)-phase rotor winding is a continuous one with no
unsymmetrical parallel paths, it is still possible to hypothesise
two series-connected conventional induction machine equivalent
circuits to represent the behaviour of the machine and, therefore,
the analysis of such machines is relatively simple.
A new approach for modelling is required when each phase of the
(p+q)-phase rotor winding consists of parallel paths such as those
7 in the 'multicircuit single-layer bar winding' . Analysis of an
SFCIM having this rotor winding has been carried out by Broadway and
7 Burbridge . This analysis is based on the application of the well
known commutator transformations to the appropriately connected
two-phase representation of the machine. The application of this
model in practice is seen to be cumbersome. The equivalent circuit
so developed is unconventional and the establishment of machine
parameters is difficult. Thus, it was required to develop a much
simpler model, to the extent that the assumptions made are not
16
different from those made in Reference 7, to deal with the SFCIM
with rotors having multicircuit arrangements.
Recently, induction motors having cage rotors with bar and
31 end-ring faults have been investigated by Williamson , using a mesh
model of the rotor. In the case of a cage induction machine with a
rotor having a faulty bar or an end-ring, both the positive- and
negative-sequence fields which are present in the airgap complicate
the analysis. Assuming no harmonic fields of the rotor couple to
the stator windings other than the fundamental positive- and the
negative-sequence fields, the fundamental supply-frequency currents
will be modulated at twice the slip-frequency depending on the
negative-sequence impedance offered by the supply system. In this
model there will be (N,+l) unknown rotor mesh currents (Nfc= number
of rotor bars in the cage) and with regard to the stator there will
be two unknowns, one representing the positive-sequence current and
another representing the negative-sequence current. With a complete
rotor, the end-ring current is zero and hence the number of rotor
mesh current unknowns reduce to Nfe. The number of stator unknown
currents reduce to one as the negative-sequence current is absent.
Inclusion of each rotor mesh current is not necessary as they form a
balanced N,-phase system. In such a situation the total number of
equations forming a mesh model of the cage induction machine would
be equal to 2. One possible way of reducing the number of equations
is to use the concept of 'cyclic' inductances which was extensively
30 used by Poloujadoff . In the model developed in Reference 31, all
currents are referred to their own sides and hence no turns ratio
17
calculations are required.
The development of the new circuit theory for the SFCIM stems
from the similarity observed between the above problem and the
manner in which the SFCIM operates. In the SFCIM, the collective
effect of the (p+q) phases of the rotor produces a positive-sequence
p-pole pair wave and a negative-sequence q-pole pair wave and these
two fields couple onto the p- and the q-pole pair stator windings
respectively. In this chapter it is shown that, in the case of an
SFCIM having a multicircuit single-layer rotor winding operating
under balanced steady-state conditions, the number of equations to
be solved in the steady-state analysis can also be considerably
reduced by the use of the concept of 'cyclic' inductances. The
model developed in this chapter is convenient to apply and the
parameters are easy to calculate. This model also does not require
the calculation of any turns ratios and all currents are referred to
their own sides. An alternative method for the derivation of
expressions for torque, which is based on the fundamental principles
of electromechanical energy conversion, is also presented.
It is also shown that the new circuit theory developed can be
conveniently applied to SFCIMs having (p+q)-phase single-layer
phase-wound rotors as well.
18
2.2 Assumptions
An agreement between the performance predicted using a
theoretical model and the corresponding measured results depends on
the accuracy of the parameters calculated, the assumptions made in
32 the calculations and on the fidelity of the model used. With
regard to conventional induction machines it has been shown that the
performance predicted using high fidelity models is much closer to
reality. Numerous factors contribute towards such high fidelity
32 models . This chapter deals with the development of a relatively
simplified model of the SFCIM. Such a model could then be used to
develop a model of any required fidelity.
The coupled circuit model presented in this chapter is based on
the simplifying assumptions which are listed below:
(a) the effects of saturation are negligible;
(b) the iron losses are negligible;
(c) the mutual inductance between any stator phase winding and
a rotor circuit is a cosinusoidal function of the rotor
position;
(d) the airgap length is short compared with the airgap
diameter and the stator and rotor surfaces are smooth;
(e) the . 2p-pole and 2q-pole stator windings are balanced with
no mutual coupling between the two windings; and
(f) the rotor winding forms a balanced (p+q)-phase system.
19
L O +» d •P
in Q>
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o L CD Cu in d .C
a .c +>
Cu in d
a i
?£ 3 CU JZ i» +» O
+> Ct- O
JZ
CU JC
+»
o
a O L CD
JQ d X JZ d a
L O +> d in
o a i
a cu
JC 0 J
1 1 / \
•fi |
JC
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C |
£ J -w+
1
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3 u
1
O 2 l-H > < EC O ta
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ta O CO
H I — I
a
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M ta
a z i — i
o
o O « OS
pa
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w o 55 h-1 CO
20
In practice, the effects of saturation on the airgap inductances can
be taken into account by adjustment using approximate saturation
factors. Unwanted space harmonics due to rotor winding are
automatically taken into account by the use of 'cyclic' inductances.
2.3 Formation of the Coupling Impedance Matrix for SFCIM having a
Multicircuit Single-layer Bar Rotor Winding
In order to develop the coupling impedance matrix, voltage
equations are established for the phase 'a' of both the 2p-pole and
2q-pole stator windings and the different meshes within the k
phase group of the (p+q)-phase rotor. These meshes are indicated in
Figure 2.1.
The self- and mutual-flux linkages within the phase windings of
either the 2p- or the 2q-pole stator winding can be easily taken
into account. In general, the (p+q)-phase rotor may consist of a
(p+q)-bar cage arrangement and the remaining slots within two
adjacent cage bars are occupied by U-shaped coils thus forming a
symmetrical rotor winding structure. At one end of the rotor, the
end-ring acts as a mechanical support while allowing circulation of
currents in the various meshes of the rotor. At the other end of
the rotor the end-ring supports only the (p+q)-bar cage structure.
The curved projecting sections of the U-loops act as the overhang of
each phase group although this is not obvious from Figure 2.1. It
is reasonable to assume that the leakage flux due to the currents in
the overhang sections of the U-loops within a phase group do not
link the nearest end-ring section of the same phase group. It is
21
also reasonable to assume that there is no mutual flux linkage
between the overhang sections of any two different phase groups, and
also between overhang sections of any phase group and the stator
windings. The mutual flux linkage between the different meshes
shown in Figure 2.1 exist primarily due to airgap fluxes. Between
two adjacent phase groups a mutual flux linkage also exists due to
the common cage bar.
Considering the phase 'a' of the 2p-pole stator winding
= i R + L -i i + M -1- (ii+ i ) + -4r yL sp dt ap sp at bp cp dt ap
(2.1)
where v = V cos (w t+ tb) (2.2) ap p p T
and the balanced set of 2p-pole stator winding currents A
i =1 cos w t ap p p
i, =1 cos (wt- 2JT/3) !> (2.3) bp p p
and
i = 1 cos (w t + 277/3) cp p P
R = resistance of a phase winding of the 2p-pole stator sp
L = self-inductance of a phase winding of the 2p-pole sp
stator
M = mutual inductance between any two phase windings of sp
the 2p-pole stator
jpr = total fundamental airgap flux linkage with phase ap
'a' of 2p-pole stator due to all rotor mesh
currents
22
<j> = an initial phase angle
t = time.
The second two terms in Equation (2.1) can be simplified by applying
the relationship i, +i = -i derived from the balanced set of bp cp ap
currents defined by Equation (2.3).
In order to evaluate the flux linkage T in Equation (2.1) it ap
is necessary to identify the total number of rotor mesh current
unknowns. Under balanced conditions this would be equal to
(z+l)x(p+q) where z is the number of U-loops within a phase group.
In the k rotor phase group shown in Figure 2.1
i , = current in the cage mesh ck
i . = current in the m U-loop located within the phase
group.
The mutual inductances between 2p-pole stator winding and the meshes
in the rotor are
M = maximum value of the fundamental mutual inductance pr
between a phase-winding of the 2p-pole stator and a
cage mesh in any phase group of the rotor
M = maximum value of the fundamental mutual inductance prm
between a phase-winding of the 2p-pole stator and
an m U-loop of any phase group of the rotor.
Therefore
P+q z p+qv
]>Mprcos(Pek) ick + ]> J
k=l m=l k=l
r v V \ F = > M cos(p0. ) i , + > > M cos(p0. i i ap Z Pr k ck L L Prm * "km
2.4)
23
The mechanical angle, 0, , is the spatial angle between a reference
axis of the phase 'a' of the 2p-pole stator winding and the axis of
the k rotor phase group indicated in Figure 2.1. This angle can
be defined as
277
9, = 9n + w t + -=d— (k-l) (2.5) k 0 m p+q
where 0n = the initial value of 0..
For all rotor meshes the 2p-pole stator produced field would
induce slip-frequency (s w ) currents. By observation of the
expression given for the rotor circuit flux linkages due to the
2p-pole stator field [Equation (2.13)], the rotor mesh currents i ,
and i . can be expressed as ukm
i , = 1 cos r
s w t - -i£E<k-i) - P e n - <j> p p p+q o ~i
i . = 1 cos ukm rm
s w t - J£E(k-l) - pen - cb - cb P P P+q 0 Tr Trm
(2.6)
(2.7)
where cb = delay angle of a cage mesh current with respect to
its flux linkage due to 2p-pole stator currents
cb = delay angle of the m U-loop current with respect 'rm
to current in the cage mesh
I = peak value of the current in a cage mesh
th I = peak value of the current in an m U-loop within rm
any phase group.
24
Following substitution of Equations (2.6) and (2.7) into
Equation (2.4), splitting the cosine product terms into cosine sum
and difference terms, and carrying out the summation from k = 1 to
(p+q); the cosine difference terms add up to zero. By applying the
frequency relationship given by Equation (1.1) of Chapter 1 to the
end result
(p+qM T = id- I cos(w t - cb ) ap 2 r p Tr'
f. (p+qM m . \ prm j. c o g ( w t _ . _ . j ( 2 g )
Z _ rm p Tr "rm m=l
Hence, the number of rotor current unknowns required for the
evaluation of Equation (2.8) is only (z+1). Let
V cos(w t + cb) be represented by the phasor V , A
I cos tot be represented by the phasor I , P P P
A
I cos (to t - (b ) be represented by the phasor I , r p Yr r
A
and I cos (« t - cb - cb ) be represented by the phasor I rm p Tr Trm rm
Applying these phasor quantities to Equation (2.8) and
Equation (2.1), the latter equation can be converted to a phasor
equation in the frequency domain w .
25
Thus
(p+q)M
P P P " P' P P' P " P « r V = R I + j w ( L - M ) I + j w EL I
1 (p+q)M V . prm T io Q\
+ > j w - — I 2.9 Z P o r m
m=l Z
where R = R ; L = L ; M = M , p sp p sp p sp
For the k cage mesh shown in Figure 2.1, the voltage equation is
0 = d-BJ + **8* + 4r*\ dt ck dt ck dt ck
+ rb(2ick" ^(k-l)" ic(k+l)) + ^ceN*
+ > r i . (2.10) Z eum ukm
m=l
where ySf = fundamental airgap flux linkage with the cage mesh ck
of the k phase group of rotor due to currents in
the 2p-pole stator winding
ysq = fundamental airgap flux linkage with the cage mesh ck
of the k phase group of rotor due to currents in
the 2q-pole stator winding ,.
r th T . - total flux linkage with the cage mesh of the k ck
phase group due to all rotor mesh currents
26
E
CU CO
•
(M •
CM
u CU U
CM
•
CM
CO 55 O l-H
H U W CO
« l-H
« o H
o « ta O CO
w u 55 <d H CO M CO
w
CM
CM
W
M ta
CM •
CM
27
r, = resistance of a bar in a cage mesh (illustrated in
Figure 2.2.1)
r = resistance of the end-ring section of a cage mesh ce
(illustrated in Figure 2.2.2)
r = resistance of the end-ring section common to a cage eum
mesh of a phase group and the m U-loop of the
same phase group (illustrated in Figure 2.2.3).
sp In Equation (2.10) the flux linkages IP f can be expressed as a
ST)ft cnn STIC*
sum of the three flux linkage components y , , TP , , y , of the
2p-pole stator winding
ySP _ yspa + yspb + ^spc ck ck ck ck
M cos(p0, ) i + M cos(pe, -277/3) i, pr k ap pr k bp
+ M cos(p0, +277/3) i (2.11) pr k cp
Simplifying Equation (2.11) using the set of 2p-pole stator currents
given by Equation (2.3)
3M Tsf = EL I cos(w t-pO (2.12) ck „ p p k
28
Using the expression for 9. given by Equation (2.5) and
Equation (1.1) of Chapter 1, Equation (2.12) can also be expressed
as
,SP ck
3M pr I cos s w t
P P ^ - f " - D - P*0 (2.13)
SQ
Similarly, the flux linkage y £ can be expressed as CK
ysq _ ysqa + ysqb + ^sqc ck ck ck ck
M cos(q9. -a) i + M cos(q0, -277/3 - fi) i, qr k r aq qr k r bq
+ M cos(qe. +277/3 -B) i qr k r cq
(2.14)
where
M qr
= an angle introduced to account for the spatial
displacement of the reference axis of phase 'a' of
the 2p-pole stator winding with respect to that of
the 2q-pole stator winding
= maximum value of the fundamental mutual inductance
between a phase winding of the 2q-pole stator and a
cage mesh of the rotor.
Invoking the frequency relationship given by Equation (1.4) of
Chapter 1, the expressions for the phase currents of the 2q-pole
stator winding become:
29
^ q = \ C O s ( - S p S q V " •2)
*bq = Jq C08<-8pBqV " 2*/3 ' •2) } (2<15)
A *
i =1 cos(-s s w t + 277/3 - cb„) cq q P q p '2'
where cb- = an initial phase angle.
Using the set of currents defined by Equation (2.15),
Equation (2.14) may be simplified as:
y 3M
sq _ qr ck 2 q P q P
I cos(-s s w t - q0. + fi - cb0) (2.16) q p q p M k ^ ~2' v '
Using Equation (1.3) of Chapter 1 and the expression for 0, given by
Equation (2.5), Equation (2.16) can be expressed as
3M A
y3? = _5L I cos ck 2 q
V^ + ^ k - D + *>n -?+ fe p p p+q (2.17)
In Equation (2.10) the flux linkage y , can be considered to
comprise several components:
- the airgap flux linkage due to the cage mesh
st current i , [1 term of Equation (2.18)]
- the mutual airgap flux linkage due to all other
mesh currents in the rotor [2 and 3 terms of
Equation (2.18)]
- the end-ring flux linkage due to the cage mesh
current and the mutual end-ring flux linkage due to
30
other U-loop currents within the cage mesh [4 and 1.TL
5 terms of Equation (2.18)]
- the slot-leakage flux of the bars of the cage mesh
[6th term of Equation (2.18)]
and assuming no overhang flux of the U-loops linking the cage mesh;
p+q p+q z
yr, = L i , + > M i +^S >M, i ck cc ck Z c cn Z Z ck,unm unm n=l n=l m=l n*k
+ > M i , +21 i , Z eum ukm ce ck
m=l
* h (2ick-ic(k-l)"ic(k+l)) (2'18)
where L = airgap inductance of a cage mesh cc
M = mutual airgap inductance between any two cage c
meshes
M = inductance of the end-ring section common to a cage eum
mesh of a phase group and the m U-loop of the
same phase group (illustrated in Figure 2.3.1)
1 = end-ring inductance of a cage mesh (illustrated in ce
Figure 2.3.2)
M , = mutual airgap inductance between the cage mesh of ck,unm
k phase group and the m U-loop of the n phase
group
31
JQ
eo
CM
CO 55
o l-H
H U
w CO
CU Ui
Cu
u CM a
eo a
CM
u 03
o E-i
O « ta O CO
w o 55
<;
n 55
3 Cu eo
a
CM
CO
CM
W
s M ta
32
L = slot-leakage inductance of a bar (illustrated in b Figure 2.3.3) .
It can he observed that Equation (2.18) contains a total number of
rotor mesh currents which is equal to (z+l)x(p+q). The number of
rotor mesh current unknowns can be reduced to (z+1) by introducing
the two rotor cyclic inductances 1 and 1 under balanced operating
conditions (Appendix 1). Referring to Equation (A1.14) of
Appendix 1
yr : 1 i , + \ l i , + > M i ck c ck Z um ukm Z eum '
ukm m=l m=l
bb ck ce ck
The inductances 1 , 1 and 1,, are c um DD
1 c
1 um
=
=
^0 a x
c g ^0
a x um
1„ = 41. sin2 *P
"bb ~ b p+q
where a - total airgap surface area
airgap surface area of a cage mesh c ^
(2.20)
(2.21)
33
airgap surface area of an m U-loop
um
g = airgap length.
Substituting for ySj\ yS£ and yr from Equations (2.13), (2.17)
and (2.19) respectively and using the Equations (2.6), (2.7) and
Equation (A1.13) of Appendix 1, Equation (2.10) can be modified to
the form
3M pr 1 d
I —pr- COS
p az 8 to t - -^P-(k-l) - P0„ P p p+q o
3M . , + —— I —JT- cos
277q S W t + flp. (k-l)+ q0n- /3+ cb„ p p p+q o T2
(r,,+2r ) + (1 +L.K+21 )-i-bb ce c bb ce dt
.1 cos r
s u> t - ! P_(k-l)-p0n-cb P P P+q 0 Tr
r + (1 + M )-tr eum um eum dt
m=l
.1 cos rm
P P p+q ' 0 Tr Trm
(2.22!
A • 2 77p
where ruu = 4r, sin . bb b p+q
Following the addition of a phase angle of _Z(k-l) + p0„
34
to the argument of all cosine functions, the frequency s w can be P P
changed to w without affecting the mathematical validity of
Equation (2.22). The previously defined phasors I , I and I and p r rm
a new phasor
I which represents I cos Q q
w t + (p+q)0Q - fi + cb2
can be employed to convert the resulting equation to a phasor
equation in the frequency domain w
3M 3M 0 = j» p r I + jto q r I
P 2 P P 2 q
r(r +2r ) bb ce + jw (1 +1.,+21 )
" p c bb ce
eum
m=l p
+ jw (1 +M ) p um eum rm
(2.23)
For the m U-loop within the k cage mesh shown in
Figure 2.1, the voltage equation is
_l_ysp + d , y s q + d , y r + r i dt ukm dt ukm dt ukm um ukm
+ r i , + eum ck
m
h=l eum ukh r , i . . (2.24)
euh ukh h=ra+l
35
where y sp ukm
„sq ukm
ukm
um
fundamental airgap flux linkage with the m U-loop
A. r^
of k phase group, due to the currents in the
2p-pole stator
fundamental airgap flux linkage with the m U-loop
of k phase group, due to the currents in the
2q-pole stator
total flux linkage with the m U-loop of k phase
group, due to all rotor mesh currents
resistance of an m U-loop excluding the end-ring
section.
Noting the form of the Equations (2.13) and (2.17), the flux
SP SQ linkages y , and y . are
ukm ukm
3M , sp prm T
y . = — - — I cos ukm n p
s u t - ^.(k-l) - P 0 n
p p P+q 0 (2.25)
3M sq qrm T
y , = — - — I cos ukm n q
277q s u t + Z^L(k-l) + q0n-/3 + <b„ p p p+q 0 r T2
(2.26)
where M = maximum value of the fundamental mutual inductance qrm
between a phase winding of the 2q-pole stator and
fr Vi
an m U-loop of any phase group of the rotor.
The flux linkage y , consists of
airgap flux linkages due to all cage mesh currents
[1st and 2nd terms of Equation (2.27)]
- airgap flux linkages due to currents in all U-loops
36
4-1* -Y*r\ +• Vt
similar to the m U-loop [3 and 4 terms of
Equation (2.27)]
- airgap flux linkages due to currents in all U-loops
dissimilar to the m U-loop [5 term of
Equation (2.27)]
- end-ring flux linkages due to all mesh currents of
the kth phase group [6th, 7th and 8th term of
Equation (2.27)]
- overhang flux linkages due to all mesh currents of
the k phase group [9 and 10 terms of
Equation (2.27)]
- slot-leakage fluxes [11 term of Equation (2.27)].
p+q p+q
yr, =^M . i + M . . i . + \ M . i ukm £ cn,ukm cn ck,ukm ck Z ukm, unm unm
n=l n=l n*k n*k
P+q z
ukm, ukm ukm Z L ukm,unh unh n=l h=l
h?tffl
m z + )M iiu + > M , i . , + M i, Z eum ukh Z euh "^h eum ck h=l h=m+l
+ 1 i . + > M u i , u + 2 1 ^ i , (2.27 oum ukm /, oum,ouh ukh b ukm
h=l h m
37
where
Mukm unm = mutual ail"gaP inductance between m U-loops of nth
and k phase groups
Lukm ukm = Self air^ap inductance of an m U-loop of any
phase group
loum = overhang inductance of an m U-loop of any phase
group
M th . = mutual overhang inductance between the m U-loop
and the h U-loop of the same phase group.
Simplification of Equation (2.27) is given in Appendix 1, which is
based on the use of 'cyclic' inductances. Referring to
Equation (A1.27) of Appendix 1
m yr. = (1 + M ) i . + \ 1 i . , + ukm um eum ck Z um u^h uh ukh
h=l h=m+l
m + > M i + > M , i ., Z eum ukh Z eu" U K ^
h=l h=m+l
+ > M .i ,. + (21,+1 )i . (2.28) Z oum,ouh ukh b oum ukm
h=l
The expressions given by Equations (2.25), (2.26), (2.28) and the
phasors 1,1,1,1 plus a new phasor * p q r rm
38
I . which represents I .cos(to t -cb -cb , )
can be used to transform Equation (2.24) to a phasor equation given
by Equation (2.29) in the frequency domain w .
3M 3M 0 = j W ELL I + joo ££1 I P 2 P P 2 q
r r + jw 1 + M
s p um eum
-^L_ + j„ (21 +1 ) s p bb oum rm
r r e u m I • / 1 , 1 1 \
+ jw (1 +M s p um eum
m
h=l rh
u r , + jw euh p h=m+l
> M u + ) 1 u Z euh Z un
h=m+l h=m+l
+ \ M
h=l h^m
oum,ouh rh (2.29,
For the phase 'a' of the 2q-pole stator winding, the voltage
equation is
aq i R + L -i-i + M -4r(i, +i ) + -ir*T (2.30) aq sq sq dt aq sq az bq cq dt aq
39
where y = total fundamental flux linkage with phase 'a' of ELQ
2q-pole stator due to all rotor mesh currents.
The form of v is similar to that of i given by Equation (2.15 aq aq
but it includes an additional phase angle 6
v = V cos(-s s w t - cb0 + 6) (2.31) aq q p q p '2 ' '
The second two terms in Equation (2.30) can be simplified using the
relationship, i = -(i, +i ), derived from the balanced set of aq bq cq
2q-pole stator currents defined by Equation (2.15).
The flux linkage y can be expressed as aq
p+q
yr = > M cos(q0, -fi) i , aq Z qr * ck k=l
p+q
+ ^ M cos(q0, -fi) i . (2.32) Z Z qrm k ukm m=l k=l
Using the expressions for i , and i , given by Equations (2.6),
(2.7) respectively and Equation (1.3) of Chapter 1, Equation (2.32)
can be simplified in a manner similar to that adopted in developing
the expression for T given by Equation (2.8). 8Lp
40
Thus
(p+q)M yr = SL i cos aq „ r
-s s u t + (p+q)0~ + cb - fi p q p ^ ^ ' 0 "r i"
(p+q)M qrm
m=l
I cos rm
•s s w t +(p+q)0„ p q p 0
+ cb + cb - fi Tr Trm r
(2.33)
Subtracting a phase angle of
(p+q)0Q- fi
from the arguments of all cosine functions in Equation (2.33) and in
expressions for i and v given by Equations (2.15) and (2.31)
respectively, Equation (2.30) can be converted to a phasor equation
in the frequency domain w . Thus P
J R q _ q I + jw (L - M ) I + jw
(p+q)M qr
s s s s q p q q q P 9 r p q p q L
1 JV p+q)M
qrm rm
(2.34!
m=l
where V represents V cos q q
wpt + cb2 - 6 + (p+q)0Q -fi (2.35)
and R = R ; L = L ; M = M q sq q sq q sq
to CO
41
CM 3 O
cq P V
& + 3 O
P. 5K
cr + P.
t<
cr SB
CT + P.
3 H "p.
3
e-4 3
Ii +
3 01
ti P.
3
CT + P,
CT S
CT + P,
3 H.
3
CM 3
ti +
.P
IH
P.
CT + P.
ti
CT
CT + P.
tt O
ti +
3 H P. 3
+
CT E I
CT IH
CT ti
CT ti
CT
« CT P.
I tl P. tl
^Pi
tl P.
-»
1 * 1 II
1 I
42
From Equations (2.9), (2.23), (2.29), and (2.34) it can be observed
that the SFCIM with two separate stator windings and a multicircuit
single-layer rotor winding can be modelled under balanced conditions
by using a (3+z)x(3+z) coupling impedance matrix.
The coupling impedance matrix and the associated voltage and
current vectors for an SFCIM having a rotor with a cage and two
U-loops (i.e. z = 2) is given by Equation (2.36). It can be
observed that the sub-matrix, which represents the coupling between
the rotor meshes, is symmetrical as expected and that the total
impedance matrix can be conveniently set up for a multicircuit rotor
with any number of U-loops per phase group. This can be considered
as a definite advantage of this model.
2.4 Evaluation of Torque
2.4.1 Asynchronous mode of operation
The steady-state torque developed by an SFCIM working in the
asynchronous mode can be established using the well known concepts
behind the expression for the torque developed by a conventional
induction machine. In the case where the 2p-pole stator is supplied
at an angular frequency of w and the 2q-pole stator is
short-circuited directly or through resistors, the commonly used
expressions for the torque are:
P T = p
P
q
r v n"wtot , v
s p
SCL ). . q tot
s s P q
q tot
s s P q
(2.37)
(2.38)
43
where T = total 2p-pole component of torque
T = total 2q-pole component of torque
(RCL). = total rotor copper losses tot
(SCL ) . = total 2q-pole stator copper losses
The rotor mesh currents and the 2q-pole stator current I q
obtained using Equation (2.36) with V = 0 and V fixed at a q P
specified value can be used to compute the total rotor and 2q-pole
stator copper losses and hence the evaluation of total torque is a
relatively simple matter. The expressions for steady-state torque
can also be developed by an alternative approach based on the
fundamental principles of electromechanical torque production.
The electromagnetic torque (T) of an electromechanical system
is given by
T
where
-d»t(i,e)| (2.39) i=constant
W'(i>0) = co-energy of the system which is a function of the f
currents (i) and position (0) of the coils.
Alternatively
[i]t 4 r tTP] (2.40)
where [i] = transpose of the current vector [i]
[y] = flux linkage vector.
sp Therefore, taking into consideration the rotor flux linkages y^j
44
*ck' ukm a n d ^ukm' t h e t o t a l instantaneous torque (T.) of the SFCIM
is given by
T. 1
(P+q) i ck
a ysP + _±_ ysq " 60, ck 60. ck k k
+ (P+q) y ukm a ysp + d ysq
m=l 60, ukm 60, ukm
(2.41)
The multiplying factor (p+q) accounts for the presence of (p+q)
phase groups of the rotor.
Defining several components of torque:
Tpi = (p+q) i . 4 - ysp
ck ck 60. ck (2.42)
T ^ = (p+q) i . * r S l ck ck 60, ck
(2.43)
and
rPi ukm
ukm
(p+q) km 4r/uL k
(p+q) \\Lrn -iejlL k
(2.44)
(2.45)
then Equation (2.41) can be expressed as
TP1 + Tqi + ck ck
V pi + Tqi Z ukr '•''•" ukm ukm m=l
(2.46)
45
sp The derivative of y r with respect to t is defined by
Equation (2.47) as
d ^sp _ 6 ^sp 6_ + 6 ysp dt ck 60, ck 6t k al ck
k
(2.47)
sp Using Equation (2.47) and the expression for f , given by
Equation (2.12) it can be shown that
d«,sp "aTck
, s w
±-vsv f- p p 60, ck I
k L p
(2.48)
Therefore, according to the definition given by Equation (2.42)
<»•«> 4ell = *£ S to .,
P P (2.49)
Taking a similar approach it is possible to show that,
tn.„\ 4 a ™sq _ Tq i
(p+q) Xck "H^ck ~ Tck
S to ^
p p (2.50)
f + \ ' d ^sp ukm dt ukm
TPi ukm
S to
P P .2.51)
and
\ • d sq ^p q ukm dt ukm
, s w . Tqi j p p ukm L
q
, s w . (2.52)
46
Applying Equations (2.49) and (2.50) in Equation (2.10)
Ip+qT rpi ck
s w ., P P + T qi
ck
S to
P P -i d yr
ck dt ck
(rKV+2r )i2
bb ce' ck - I ck
r i i
eum ukm (2.53;
and applying Equations (2.51) and (2.52) in Equation (2.24)
1 Tp+qT ukm
s w P - ] + Tqi
J ukm
s w .> i
P P d „,r .2 ukm dt ukm um ukm
m
eum ck ukm ukm 2. e u m u^h h=l
ukm Z euh u^h h=m+l
(2.54
Equation (2.53) contains terms representing the instantaneous
Pi ck
torque due to all cage meshes in the rotor. Average values for T
and T . can be established by averaging the right hand side.
Equation (2.54) represents the instantaneous torque due to the m
U-loops within all cage meshes only and a summation of both left and
right hand sides is required to represent the total instantaneous
z z
torque components \ T , and \ T , . This process, followed by
m=l m=l
47
d r averaging, would reveal that all terms due to i ,—Ly , in
ck dt ck z Equation (2.53) and all terms due to ) i , —dP1", in
£, ukm dt ukm m=l Equation (2.54) when collected together add up to zero. Such a
result would be expected since there will be no average torque
produced due to interaction between rotor currents and rotor flux
linkages due to rotor currents. The terms which remain would
represent the copper losses within all rotor circuits. In the end
result the average value of both components, T and T of the total P q
torque would appear. However, in order to evaluate the components
separately and also the total torque, another equation is required.
A choice is made here to develop an alternative expression for the
2q-pole torque component T . The instantaneous value of the total
2q-pole torque component T is given by
,qi 3i sq aq 60, aq
(2.55)
The factor 3 accounts for the torque due to the three phases of the
2q-pole stator.
It is possible to demonstrate that
d yr dt aq
s s w p q p y
60k aq (2.56)
Therefore
3i - i / aq dt aq
s S to p q p rqi
sq (2.57)
48
Applying the relationship given by Equation (2.57) in
Equation (2.30) with v set to zero aq
3i R + i (L - M )JLi aq aq aq' sq sq'HTF aq
r S S to ., p q p
L q J
Tqi sq
(2.58)
Upon averaging, Equation (2.58) reduces to
I R q T = 3 —3- q -q 2 s s to
p q p
(2.59)
The detailed steps involved in the simplification of
Equations (2.53) and (2.54) for a case where the multicircuit
single-layer bar rotor winding has two U-loops are given in
Appendix 2.
2.4.2 Synchronous mode of operation
When the 2q-pole stator side is fed from an a.c. source the
machine may assume synchronous operation and in such a situation the
torque T will be given by
R = 3
V I -3-iicoBfi (2.60)
S S to
p q P
49
a 55 I — I
a 55
o H
o w Q 55 > O I
w CO w CO
< a cx, i
CM
W
o ta
50
2.5 Formation of the Coupling Impedance Matrix for SFCIM having a
Single-layer Phase-wound Rotor Winding
From the structure of the steady-state coupled circuit model
described by Equation (2.36), it is possible to establish the
corresponding matrix equation for an SFCIM having a (p+q)-phase,
phase-wound rotor of the form given in Figure 2.4.
The unknown currents are now, I , I and I only, where I is p q r r
the current in the winding forming a phase group in the rotor.
Thus, the order of the impedance matrix is 3x3. The corresponding
matrix is given in Equation (2.61). The elements of this impedance
matrix can be derived by examination of the steps involved in the
derivation of Equation (2.36). In Equation (2.61) the parameters of
concern are
M = maximum value of the fundamental mutual inductance pr
between a phase winding of the 2p-pole stator and a
phase winding of the rotor
M = maximum value of the fundamental mutual inductance qr
between a phase winding of the 2q-pole stator and a
phase winding of the rotor
L = airgap inductance of a phase winding of the rotor cc
M = mutual airgap inductance between any two phase
windings of the rotor
1, = total slot-leakage inductance of a phase winding b
1 = total overhang leakage inductance of a phase o
winding
r, = total series resistance of a phase winding. b
CO
CM
o H
51
.ft
a1
+ p. ft
3
a*
+ ft
ft
o o ft 3
ft
as
ft C*4
ft
Pi o ft
S I
ft
ft 3
P5 ft
co CM
ft
ft CO
52
2.6 Application of the Circuit Theory of SFCIM
2.6.1 Introduction
The results of the application of steady-state circuit theory
to an asynchronously operating SFCIM are discussed in this section.
Comparisons are made with the experimental results wherever
possible.
An overall comparison of theoretical and experimental results
is not made but to justify the validity and the accuracy of the
model, comparison is made between current levels and the torque
produced.
2.6.2 Description of the experimental SFCIM
The experimental machine is designed to function as a
(2+6)-pole SFCIM, thus giving the highest possible cascade
synchronous speed for a given supply frequency. The 36-slot stator
frame consists of independent 2-pole and 6-pole, star-connected,
balanced, three-phase windings (Photograph of Figure 2.5). The
44-slot rotors were totally interchangeble. This, however, does not
permit the comparison of different rotor winding configurations as
the stator windings have to be, in general, designed to match the
rotor windings. Design values of the peak 2-pole airgap flux
density and the 6-pole airgap flux density are 0.2T and 0.3T
respectively. The corresponding stator rated voltages are 140V and
50V (line to neutral) respectively.
Figure 2.6.1 shows a 4-phase multicircuit rotor, which consists
of five U-loops per phase group, thus leaving one slot unoccupied.
The four bar cage was totally excluded from the winding due to
53
FIGURE 2.5 PHOTOGRAPH OF (2+6)-POLE STATOR FRAME
54
2.6.1 MULTICIRCUIT SINGLE LAYER BAR WINDING
2.6.2 PHASE-WOUND SINGLE LAYER BAR WINDING
FIGURE 2.6 PHOTOGRAPHS OF 4-PHASE ROTOR WINDINGS
55
difficulties in construction. The form of the phase-wound rotor
illustrated in Figure 2.4 and by the photograph of Figure 2.6.2 is
attractive due to its simplicity in construction. This, however,
does not represent an optimised design. Some of the design aspects
of this form of rotor winding will be examined in Chapter 5.
2.6.3 SFCIM having a multicircuit single-layer bar rotor winding
2.6.3.1 The coupling impedances
Many of the elements of the coupling impedance matrix can be
33 34 calculated using established methods ' . The expressions for the
rotor 'cyclic' inductances required have been given earlier in the
appropriate sections. The inductance (L - M ) or (L - M ) can be P P q q
considered to be the sum of the fundamental magnetising inductance
31 and the total leakage inductance of the stator winding of concern
Of particular importance are the inductances M and M (also M r prm qrm pr
and M if a cage mesh is present in the rotor) which can be qr
calculated by a knowledge of the fundamental magnetising reactances.
The inductance M is given by prm
H =d_J_X prm 3 mp
, sin pa /l m
N p pse
k (2.62) sp
where
X = 2p-pole fundamental magnetising reactance referred mp to the 2p-pole stator side
56
coil pitch of an m U-loop in mechanical radians
effective number of turns per phase of the 2p-pole
stator winding
2p-pole skew factor.
The expression for the inductance M can be obtained by replacing qrm
the suffix 'p' by 'q' in the Equation (2.62). These inductances and
other airgap inductances have to be corrected for the effects of
magnetic saturation by the use of appropriate saturation factors.
These factors can be calculated by considering the mean length of
the flux paths corresponding to the 2- and the 6-pole flux density
distributions and the magnetic characteristic curves of the steel
used. A more rigorous approach (described in Reference 33) was
employed in the present work. The rotor airgap inductances were
also corrected using an approximate saturation factor calculated by
the consideration of mean length of flux paths from one phase group
to another. In a four-phase rotor there will be two such lengths
and hence two saturation factors can be calculated. The average
value of the two was considered to be accurate enough to represent
the effect of saturation on the rotor airgap inductances. As
expected this value lay between the 2-pole and the 6-pole saturation
factors.
In the calculation of bar inductance 1, and resistance r, , the b b
skin effect on the rotor bar sections embedded in the slots can be
incorporated by employing the ladder network approach described in
Reference 33. However, the effective resistance and inductance of a
a m
pse
sp
57
bar calculated at a frequency of 50Hz for the available depth of the
rotor slots were little different from those values calculated for
direct current. Therefore, the skin effect was not taken into
account in the theoretical calculations.
2.6.3.2 Current/torque versus slip curves
The experimental testing of the SFCIM was carried out using a
d.c. dynamometer. The torque was measured using a d.c. bridge
having strain gauges mounted on a restraining arm attached to the
test machine mounted on trunnion bearings. Although these
measurements have been corrected for the friction and windage
effects, the hysteresis of, and the temperature effects on, the
strain gauges cannot be easily eliminated. However, care was taken
to minimise these experimental errors.
Figure 2.7 shows the calculated and measured variation of the
2- and 6-pole stator current with slip (s = s s ) when the machine P q
is supplied on the 2-pole stator side; while Figure 2.8 shows the
corresponding 2-pole stator current loci. Figure 2.9 shows the
current levels on the two stator sides when the 6-pole stator side
is supplied; while Figure 2.10 shows the corresponding 6-pole stator
current loci.
The calculated and measured variation of the torque with slip
in the two cases considered above is shown in Figure 2.11. At
standstill the measured torque was not the same at different rotor
positions and hence the value indicated is only an average value.
58
s
cn
ex, r-i •J CO
rc H r—1 S
H 52 W Pi « 3 o ta O
2 O l-H E-> < \—I « < >
N
ac o in -P aj
. . 55 • 1
-»—-> o •=r , — i
+-> cd tj 0) •H ,-H ft ft 3 01
EH
o +J «3 -p cn
OJ r—1
O ft 1
C*J
•vs <u •H 3 O PH • H
o •P
fH O Si cn
M O P cd P cn cu r-H
o ft 1 CD
CM
W
o M ta
(H) lNByyno
59
m
0
CD *J
IO
3 U
IO
u
[ I I I
0
I
0
I
0
1 1
0
1
0
\ ° > \
0
1 1
a u 3 W a CO
E 0
1 1
—
—
—
—
—
—
—
—
1 1
m
cu
CJ
ts
CE
r—1
.—1
cn •
oo
r-
CO •
IT) a
"tf
co •
2: LU CC cc ZD CJ
az a i— cx i— CO
LU _l CD CL. 1 CM W J
CO
CC
o o
2 W Pi PH
o <
CO
w o PH
1 C<1
00 •
M
H B ta
in Tf m c\J «-< c a O T C D r - t o L O ^ t - c o o j CS
[ (tn iN3yyno y o m s Biod-a}1^
60
is
Q_
CO
PH l-H J K)
EC H hH i£
H 2 W P3 PH
13 u PH
O
2 O hH H < h-1 PH
< >
a> •
CM
w
B O M UH
N CC O LO
p cd , . 2 1 1
_3 ^--> in LO
•P cd
id OJ •H ,—1 ft ft 3 cn tn O P cd •p CO
cu ,—1 o ft 1 to
T=l 01 p •H 3 O CH • H
O 4-> tn O si cn CH
O •P
cd p cn QJ rH
o ft 1 CM
in •
"<*
S3 I
"«*•
un •
OD
S a
ro
in .
OJ
si •
ru
in ta
(UJ iN3yyn3
61
-a CO
L 3 W a CD
£ 4
< <
tS •
•<*•
in a
cn
ts
in
ca
ts •
TC
in a
m
ts •
oo
in •
CM
ts a
CM
m ts in ts
CE
m
in •
OJ
ts •
ro
m a
»—1
a
—'
^ UJ DC CC ID CJ
CC CD
CE h-CO
LU _] CD Q_ 1
CD w->
CO
DC
u o J E-i
2 P3 P3 53 C_>
o < H Kl W
o ft | to
o F—1
CM
W
3 C5 H P-
{(B)iN3yyn3 yoidis aiod-gj^i
SI
62
-a <D
—
Q. Q-3 M
C
o •p
10 •P (0
co •
o tx
CO
-o CO t-3 W
a CO
E
1
1 *
\ *
•o \ <D \ *> ^ \ to \ — \ 3 \ O ' — (0
o
- l - « —
4 0
\ e
1 \
ss^s^« 04
\ 0
1
-a CD
•P
IO
3 U
a o
-e
-o CD
L 3
« 10 a> E 0
\l
~ C ^
-tJ CO
Q.
a. 3 W
c. o •P 10 •»
w
CD
_ o a.
C\J
1
—
—
—
" ™
—
1
CM
CO
in
co
oo
cn
S3
cn S
CO
cc E-i
W
P3 O H PH
o 2 O < hH
P5 >
CM
W
O M PH
Ul a
tn
S3 •
CO m •
OJ
ts •
OJ
in ts in si
(WM 3noyoi
63
The theoretical and experimental magnitudes of the current
levels in all cases are seen to be in close agreement despite the
simplifying assumptions made in the development of the coupled
circuit model. However, the calculated and measured stator current
loci are not seen to be in close agreement. This has been the case
in using simple models in predicting the performance of conventional
32 induction machines . With regard to conventional induction
machines, it is reported that the inclusion of the corrections due
to factors which are generally ignored in simple models, lead to
32 more acceptable theoretical predictions . Among these factors are
the effects due to unwanted harmonics, variable degree of saturation
of main flux paths and leakage flux paths, skin effect on the bar
type conductors and the iron losses.
It follows, therefore, that at high slips the saturation of the
leakage flux paths generally lead to higher torque levels than those
predicted using simpler models which do not account for this. The
skin effect on rotor bar conductors also contributes to a higher
starting torque than that predicted but in the test machine this
contribution is negligible. It can therefore be concluded that the
saturation of the leakage reactances cause the test machine to
develop a higher starting torque than that predicted theoretically.
Theoretical study on the harmonic behaviour of the multicircuit
rotor winding at different speeds reveals that the percentage
magnitude of each space harmonic mmf varies as the speed changes
(Appendix 4). This behaviour has been noted by Broadway and
7 Burbridge . It is due to the presence of separate U-loops within
64
each phase group in which the distribution of currents can vary with
the speed. In practice, this behaviour can be expected to lead to a
varying ratio between the 2- and 6-pole flux density. As a result
the degree of saturation also changes with the speed of the machine;
an effect which cannot be simply taken into account in theoretical
modelling. The saturation of the various flux paths in the SFCIM is
a complex problem due to the simultaneous presence of the 2- and
6-pole fields. This is further aggravated by the presence of other
unwanted rotor harmonic fields. This can be considered as an
important contributing factor for the differences observed between
the theoretically calculated and measured quantities.
2.6.4 SFCIN having a single-layer phase-wound rotor winding
2.6.4.1 The coupling impedances
The elements of the coupling impedances in Equation (2.61) can
be calculated in a similar manner outlined in Section 2.6.3.1. The
inductance M is given by pr
N , k M --J--JL X pr 3 u mp
wpr
N pse
k (2.63) sp
where k = 2p-pole winding factor of a phase winding of the wpr
rotor
N = number of turns of a phase winding of the rotor. r
The inductance M can be obtained by replacing the suffix 'p' qr
by 'q' in Equation (2.63).
65
2.6.4.2 Current/torque versus slip curves
Figure 2.12 shows the variation of current with slip in the
2- and the 6-pole stator windings when the 6-pole stator side is
supplied while 2-pole stator is short circuited. The corresponding
measured and calculated 6-pole stator current loci are shown in
Figure 2.13. The variation of the torque with slip for the same
case is illustrated in Figure 2.14. As was stated before, the
performance of the SFCIM having a phase-wound rotor can also be
examined using two series connected equivalent circuits of
conventional induction machines. This equivalent circuit is given
in Figure 3.1 of Chapter 3 of which the parameters can be calculated
33 34 using established methods ' . Figure 3.10 of Chapter 3 gives the
magnitudes of these parameters. The calculated results using this
equivalent circuit are also illustrated in Figures 2.12, 2.13
and 2,14.
The theoretical predictions using the coupled circuit model and
the equivalent circuit can be observed to be in good agreement.
Minor differences between the two can be attributed to the accuracy
of the parameters calculated. A reasonable agreement can be
observed between the measured and the theoretically predicted
quantities. In the case of the phase-wound rotor winding, as rotor
winding factors can be calculated with respect to the 2- and 6-pole
7
fields, the ratio of the 2- and 6-pole airgap flux densities is
nearly a constant and hence the degree of saturation due to main
flux can be assumed to stay constant. Although this is the case
66
ts
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69
with respect to the saturation of the magnetising reactances of the
machine (X and X ), the saturation of the leakage flux paths can
still lead to unpredicted variations in the current levels.
2.7 Sunary and Conclusions
A steady-state circuit theory for the SFCIM has been developed
starting with the first principles. The seemingly complicated mesh
configuration of the multicircuit rotor winding is mathematically
represented in a much simpler form employing the concept of 'cyclic'
inductances, under balanced operating conditions. The phasor
equations established are similar to those which are usually found
in the mesh analysis of electrical circuits. For an SFCIM having a
multicircuit rotor winding with any number of U-loops per phase
group, the coupling impedance matrix can be established directly by
observation of a previously established coupling impedance matrix
such as that given by Equation (2.36).
The theoretically predicted results are in reasonable agreement
with the experimental results despite the simplifying assumptions
made in the development of the coupled circuit model. Saturation of
the machine can be considered as an important factor which
contributes towards any differences. A separate investigation is
required, therefore, to examine the effects of saturation on the
theoretical model.
Unlike the case of the phase-wound rotor winding, a specific
winding factor cannot be assigned to a phase group of a multicircuit
rotor winding due to the presence of separate circuits where the
currents are distributed differently at different speeds. As a
70
result, direct access to the magnitude and phase angle of these
currents will be of considerable use in the design. The
distribution of currents in the various U-loops is also of use in
examining the harmonic behaviour of the rotor winding.
The coupled circuit theory developed in this chapter, for an
SFCIM having a multicircuit rotor winding, does not lead to the
development of an equivalent circuit. In Reference 7 Broadway and
Burbridge presented an equivalent circuit for an SFCIM having a
multicircuit winding. The complexity of their equivalent circuit
arises from the U-loops which form the parallel paths in each phase
group and as a result the development of procedures for the
experimental determination of its parameters seems to be difficult
to achieve.
The calculation of the parameters of the simple equivalent
circuit (Figure 3.1 of Chapter 3) of an SFCIM having a phase-wound
rotor is relatively straightforward. However, the presence of an
extra loop in this circuit makes the experimental determination of
its parameters difficult. Chapter 3 deals with suitable
experimental procedures which can be used for the estimation of
these parameters.
71
CHAPTER 3: EXPERIMENTAL DETERMINATION OF PARAMETERS OF THE
EQUIVALENT CIRCUIT OF AN SFCIM HAVING A PHASE-WOUND
ROTOR
3.1 Introduction
The per phase equivalent circuit of an SFCIM having a
phase-wound rotor is shown in Figure 3.1. In forming this circuit
it is generally assumed that the coupling between the rotor and the
two different stator windings exists only via the 2p- and the
2q-pole mmf waves respectively and that the machine is operating
under balanced sinusoidal conditions.
sp J* sp
r /s Jx Jx rp p rp rq
r /s rq p
Jx. r /s s sq p q
2p-pole machine 4 } 2q-pole machine
FIGURE 3.1 PER PHASE EQUIVALENT CIRCUIT OF AN SFCIM HAVING A
PHASE-WOUND ROTOR
With reference to the above circuit
r , r' = resistances of 2p- and 2q-pole stator windings sp sq
72
and
x , x' = leakage reactances of 2p- and 2q-pole stator o ]J S CJ
windings
r' , r' = resistances of 2p- and 2q-pole rotor winding
components
x' , x' = leakage reactances of 2p- and 2q-pole rotor winding rp rq
components
X , X' = 2p- and 2q-pole component magnetising reactances mp mq
R , R' = 2p- and 2q- pole component core loss resistances. cp cq
All reactances are evaluated at the angular frequency w and the P
primed symbols represent parameters referred to the 2p-pole stator
side.
Although, in general, the parameters in the equivalent circuit
shown in Figure 3.1 are assumed to be constant, they are subject to
variation with operating conditions. Besides the variation of the
magnetising reactances with the flux level and the leakage
reactances with current level, the resistances representing the iron
losses are frequency dependent. In the SFCIM, the rotor current
frequency is [Equation (1.6) of Chapter 1] higher than usual and for
this reason the rotor iron losses cannot be ignored. A detailed
discussion on iron losses in SFCIMs is given in Chapter 4 which
examines the manner in which they vary over a range of speeds. It
is evident, therefore, that the equivalent circuit given in
Figure 3.1 is only an approximate representation. However, if the
magnitude of iron losses is small compared with the power rating of
the machine, which is usually the case, the inaccuracy involved in
73
the parameters representing the iron losses would not play a
significant role in the estimation of the performance using the
equivalent circuit in Figure 3.1.
25 From the available literature it is clear, that no
experimental technique is available to determine the parameters of
the equivalent circuits for SFCIMs with phase-wound rotors. As a
result, in much of the work relating to SFCIMs with phase-wound
rotors, the experimental results have been supplemented only by the
calculated results '
In this chapter, a novel technique suitable for the
experimental determination of parameters of the equivalent circuit
of an SFCIM having a phase-wound rotor is described. Experimental
results from the tests carried out are presented together with the
theoretical predictions.
Unlike the case for conventional induction machines, the
equivalent circuit given in Figure 3.1 has three sections of
interest; namely: the 2p-pole stator; the composite rotor; and the
2q-pole stator. Pertaining to the calculation of the equivalent
circuit parameters using the experimental results, a fundamental
difficulty arises in splitting the experimentally determined total
leakage reactance and assigning it to the three sections. If the
design details are known, the splitting becomes relatively simple.
In the case of the conventional induction machine this poses no
difficulty, as well known relationships are available for the
separation of the total leakage reactance into the relevant
components of the stator and rotor. This chapter also investigates
74
how the performance would be affected by different distributions of
the total leakage reactance in various sections of the equivalent
circuit.
3.2 Experimental Determination of Equivalent Circuit Parameters
3.2.1 Introduction
For a cascade connected system consisting of two similar
17 induction machines, Smith applied a novel but a simple technique
to determine their parameters. These parameters were also measured
using the conventional technique applied to induction machines and
good agreement has been reached between the results of the two
techniques. In the former technique, when the two stators are
supplied at rated voltages at standstill, there are rotor angular
positions at which no rotor currents flow. These positions are thus
equivalent to the no load situation of each machine and the torque
on the rotor is zero. The input real power, current and applied
voltage are then used to estimate the shunt branch of the induction
machine's equivalent circuit which represents the magnetising
current and the core loss. With proportionately reduced voltages
applied to both stator windings, if the rotor is turned by a certain
angle, an unstable position is reached and short circuit conditions
are exhibited by each machine, thus enabling the series parameters
to be calculated. The angle by which the rotor should be turned can
be best explained if two back-to-back connected windings are
considered. In one position, the instantaneous voltages on the two
windings oppose each other exhibiting the no-current situation,
75
while in another position they aid each other, so exhibiting the
short-circuited situation. The techniques developed with regard to
the experimental determination of parameters of the equivalent
circuit given in Figure 3.1 are based on the above techniques.
3.2.2 Determination of shunt parameters
In this section, the theory behind the experimental
determination of the shunt parameters X , R , X' (X ) and R' mp cp mq mq cq
(R ) is given. cq
While the SFCIM is doubly-fed at the same frequency of w at P
standstill the rotor can be manually turned to (p+q) positions per
revolution, at each of which no torque is exhibited on the rotor.
At these positions, both the 2p- and 2q-pole stator current levels,
while maintaining their balance, can also be found to be small
compared with the levels observed at other rotor positions. At or
near these positions the stator currents mainly correspond to
magnetisation and core loss. Thus, the voltage drop in the series
impedance section consisting of winding resistance and leakage
reactance of both stator windings can be ignored and the circuit of
interest would be as shown in Figure 3.2.
FIGURE 3.2 EQUIVALENT CIRCUIT TO ESTIMATE THE SHUNT PARAMETERS
76
where
V =V | 0 = 2p-pole stator supply voltage
V'=V'|6 = 2q-pole stator supply voltage (V ) referred to the
2p-pole side
6 = phase difference between the voltages V and V p q r' = r' + r'
r rp rq x' = x' + x' .
r rp rq
If P = per phase real power input measured on the 2p-pole JT*
stator excluding the 2p-pole stator copper loss
and P = per phase real power input measured on the 2q-pole
stator excluding the 2q-pole stator copper loss
the total real power input (P ) excluding the stator copper losses
is given by
P. . = P + P tot p q
V2 V'2
P-+ _ i + |I' |2 r' (3.1) R' cp cq R R' r r
where I' = per phase rotor current referred to the 2p-pole
stator side
and is given by
V | 0 - V | 5 n I n I I» = P' q (3.2
r IZ;I IJL
where Z^ = rj. + j x'r = |Z^| |_e_
77
From Equation (3.1) and (3.2)
aP. . 2V V sin 5 tot p q ,
" - ^ (3.3) as |z;|Vr r
Therefore, for any combination of magnitudes of V and V (V ), P q q
the minimum total real power (P ) flows when 6 = 0 . The angle 5
can be varied by changing the phase of one supply with respect to
the other or by turning the rotor to different positions.
For 6=0, if the voltage V (V ) is varied while keeping the
voltage V constant, then
tot
av' q
tot
av» q 6 = 0
2V 2(v - V) -i - P * (3.4) R' |Z» T/r' cq ' r' r
To find the minimum value P , by equating the right hand side of
Equation (3.4) to zero
,.2 V |Z' |7r' _P = LL! L + 1 (3.5) V R' q cq
78
2 If R' is very much larger than |Z' | /r', which is usually the case, cq
then
V p * V q (3.6)
and according to Equation (3.2) the rotor current I' (I ) should be r r
zero. Thus, the total core loss (P ), which is the minimum value
n .m . . ,
of P is given by
C t = v - f — • — i «3'" 1 1
[ — • — ] 1 R R' J cp cq
By plotting the variation of P. , with 6 observed in the
vicinity of 6 = 0, a set of curves could be obtained for various
values of V (V ) and by selecting the minimum value of P. . from q q ~ tot
the set of curves, the value of P. . can be found. This means that, tot
it is experimentally possible to detect the condition described by
Equation (3.6). At this point, the voltage, current and real power
readings corrected for the copper losses in the stator windings can
be used to estimate the parameters X , R , X' (X ) and R' (R ). mp cp mq mq cq cq
The terminal voltages measured in this case can be used to calculate
the effective turns ratio between the 2p- and the 2q-pole stator
windings. Hence, the calculated parameters can be referred to any
desired stator side.
79
The above ratio of voltages and the current levels under the
prevailing conditions of the test described in this section can also
be calculated theoretically, using the matrix Equation (2.61) of
Chapter 2. Using this equation it can be readily shown that
R + ju (L - M ) P P P P R + jw (L - M )
L q P q q
(3.8)
and M qr (3.9) M pr
It should be noted that the coupled circuit model developed in
Chapter 2 ignores the presence of iron losses in the machine and
thus, Equations (3.8) and (3.9) are only approximate.
3.2.3 Determination of series parameters
In this section, the theory behind the experimental
determination of the series parameters of the equivalent circuit of
Figure 3.1 is given.
While the SFCIM is doubly-fed at the same frequency of w , the
rotor can be turned to a position halfway between any adjacent two
of the (p+q) positions obtained in the test described in
Section 3.2.2. In this case the stator voltages would have to be
considerably reduced in order to lower the current levels to safe
values and the rotor has to be locked to keep it in position. It is
generally desirable that the ratio of the applied voltages on the
80
two stator windings be maintained approximately equal to the
effective turns ratio which was obtained in the previous test.
The prevailing terminal conditions are similar to those in the
case of the short circuit test of conventional induction machines
and therefore it is reasonable to assume that the current levels
observed on the two stator windings do not contain components of
currents due to core losses and magnetisation. Figure 3.3 shows the
circuit required for the estimation of series parameters.
sp
/N
V|0. P
Jx. sp Jx, r
i \=i PH*
">
JX sq sq
/N
V q
FIGURE 3.3 EQUIVALENT CIRCUIT TO ESTIMATE THE SERIES PARAMETERS
For Figure 3.3
VPI2_ = iplii. l
ztll!L+ \\J- (3.10)
where
I =1 |-cb = 2p-pole stator current per phase P P
cb = phase angle of the current I with respect to the
2p-pole stator voltage V
81
and therefore
<rc«
+ r^. + r I J + J (x + x' + x» ) sp r sq' ° sp r sq
R + j X
lZtl Lfl
[ IZ. P1 t
( V2 + V'2 - 2 V V cos 6 ) 1 / 2
P q P q (3.11)
According to Equation (3.11), in the vicinity of 6 = n, while
keeping the magnitudes of the two voltages constant and by adjusting
the position of the rotor it is possible to obtain the maximum level
of the current I . In effect this means that both stator current P
levels would reach the maximum level simultaneously. The phasor
diagram in Figure 3.4 illustrates this situation.
FIGURE 3.4 PHASOR DIAGRAM UNDER MAXIMUM STATOR CURRENT LEVEL
CONDITION
82
V
The voltages V and V* can also be expressed as P q
Vp = (R1 + jXj) Ip (3.12)
Vq = "<R2 + JX2> *p <3'13>
where
(Rj+ R2) + j (Xx+ X2) = (R +j X)
It can be considered that the partitioning of the resistance R and
the reactance X into the components R,, R', X,, and X' correspond to
a fictitious short-circuit as illustrated in Figure 3.5.
JX, 1 ® JX, a .nom.
R,
/N /N
fictitious
short-circuit V
® FIGURE 3.5 CIRCUIT SHOWING THE FICTITIOUS SHORT CIRCUIT
This short-circuit, in effect, indicates the equipotential nature of
the points marked A and B. From the voltage, current and real power
readings it is now possible to calculate the partitioned parameters
R , X.. , R' (R9) and X' (X9). These parameters can be referred to
any desired stator side using the effective turns ratio obtained in
Section 3.2.2. As the stator winding resistances are directly
83
measurable, a value for r' can be calculated but the calculated r
leakage reactances X. and X' cannot be directly separated into their
components x , x' and x' . In this test, according to the sp r sq
Equations (3.12) and (3.13), the ratio in which the total series
resistance and total leakage reactance are shared by the two
circuits on either side of the fictitious short-circuit depends on
the ratio of the voltages applied on the two stator windings. It is
theoretically possible to infer a situation where the fictitious
short-circuit appears across the input terminals of one stator side
of the machine amounting to zero applied voltage on that stator. In
which case the applied voltage on the other stator winding would
have to be considerably raised in order to maintain a reasonable
current level. It is advisable, however, to maintain both voltages
reasonably low so that the errors due to core losses and
magnetisation are minimised and the fictitious short-circuit is not
moved too far towards the input terminals of the equivalent circuit.
The ratio of the two stator current levels obtained in this
test should be approximately equal to the effective turns ratio
obtained in Section 3.2.2.
3.2.4 Experimental results
3.2.4.1 Shunt parameters
The experimental work was carried out on the SFCIM having the
phase-wound rotor which was described in Section 2.6.2 of Chapter 2.
The test rig included three-phase power measurements on both stator
sides supplied via variacs.
84
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86
It was considered useful that the rotor phase winding current
levels, in the vicinity of 6 = 0, be observed experimentally. For
this purpose, all rotor phase windings were open circuited and
connections were brought out using carefully prepared jumper leads.
The currents through these external loops were observed using a Hall
effect current probe connected to an oscilloscope. It was decided
that the magnitude of 2-pole stator voltage V (p = 1) be kept
constant and the magnitude of the 6-pole stator voltage V (q = 3)
be varied in subsequent tests. In the vicinity of 6 = 0 and
V = V, as expected, all rotor mesh currents contained harmonic P q
currents having negligible amplitudes but did not show any
components corresponding to the stator supply-frequency. It was
also noticed that not all four phase current magnitudes reached the
near zero level simultaneously. All magnitudes were sufficiently
small, however, and their effect on the overall result was assumed
to be negligible.
To estimate the shunt parameters it was necessary to record the
real power flows P (p=l), P (q=3) and the corresponding
voltage and current readings in the vicinity of 6 = 0. Figure 3.6
illustrates the variation of P1 and P„ with rotor angular position,
for different 6-pole stator voltages while the 2-pole stator voltage
was maintained constant at a nominal value of 140V. The rotor
angular position was measured with respect to an arbitrary point on
the stator frame. Adjacent to each curve the corresponding mean
value of the three 6-pole stator line-to-neutral voltages is given
and the falling curves represent P1 and the rising curves represent
P„. The variations in the volt-ampere flows are illustrated in
87
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88
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89
Figure 3.7. In this case the falling curves correspond to the
6-pole stator side volt-ampere flows (S„) and the rising curves
correspond to those (S1) of the 2-pole stator.
Using the results of Figure 3.6 it is possible to plot the
variation of Ptot (Ptot= Pi+P3)
with the rotor angular position for
the different 6-pole stator voltages chosen. These curves are shown
in Figure 3.8. It is evident that, although the variation of P. tot
with the rotor position is small, minimum points clearly exist in
all the curves. Considerable effort was required to obtain
consistent results as the meter readings were sensitive to the
applied voltages on the two stator windings. Due to the nearby
industrial environment, voltage fluctuations on the supply system
were inevitable. As can be observed, the doubtful points on this
plot have been omitted or given less significance in preparing the
curves. It can be seen that minimum power level P. of each curve tot
occurs at nearly the same rotor angular position indicated by the
dotted vertical line; an occurrence which agrees with
Equation (3.3). From the curves in Figure 3.8 it is possible to
select the values of P. for each curve. The plot of P. versus tot tot
the corresponding 6-pole stator voltage is given in Figure 3.9.
From this figure it is possible to obtain the value of the 6-pole
stator voltage at which the minimum value of P. . occurs. This
situation is clearly described by Equation (3.6), To obtain the
individual P., and P„ corresponding to this voltage, the curves given
in Figures 3.6 and 3.7 require interpolation. These interpolated
curves are shown by dotted lines. The values of P1 and P„ thus
90
obtained can be used to calculate the resistances representing the
iron losses on either pole pair machine component. Using the
interpolated values of S-. and S„ and the estimated values of the
above resistances, the magnetising reactances can be calculated.
These parameters are
X , = 72812 X „ = 56.812 ml m3 R . = 852212 R „ = 661fi cl c3
The experimental value of the effective turns ratio between the
2-pole and the 6-pole stator windings is equal to 2.66. Using this
ratio, the parameters can be referred to any stator side. Hence,
the shunt parameters of the 2-pole side when referred to the 6-pole
side are
R', = 1204fi X' = 103f2 cl ml
Using Equations (3.8) and (3.9), the calculated value of the
ratio of the 2- to 6-pole stator voltages was found to be equal
to 2.58. The calculated and experimental values of the ratio of 2-
to 6-pole stator current levels were found to be approximately the
same.
3.2.4.2 Series parameters
The jumper leads which had no influence on the previous test
were removed and the original rotor connections were restored. The
stator voltages were appropriately lowered and the rotor shaft was
91
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CO
oo •
CO l-H
a> ai
•
t-
i—<
«=f
• o to •
CO r-i
• t-co
1—1
e-• to
CO t-
• CM
r-l 00
• CM
»* to
• CM
r—1
• *
CT>
' o
o CM
* r—1
CO
a> CO LO
CO
+-> cn <v E-i
CO w 1—1
03 w CO
w 33 H W z t — 1
s OH
W H W Q
O H Q W s 03 o OH
03 oa OH
H CO 05 H W 33 H
S o 03 OH
CO H •J
t= CO 03 03
CO 03 W H OI s < 03 < OH
92
turned by approximately 45 mechanically. It was noted that the
2-pole and 6-pole stator currents do not reach their respective
maximum values simultaneously when the rotor is turned. Between the
maximum readings of the two stator current levels, however, the
power readings on both stator windings did not show a significant
variation. Also, none of the maximum levels of stator currents
observed varied significantly when the other stator current
approached its maximum. Therefore, the error introduced into the
test results by ignoring the presence of the finite shunt impedances
appears not to be of particular significance.
It was considered useful to perform the test on the two stator
windings at different supply voltages. This would enable
examination of the dependence of the calculated parameters on the
different stator supply voltages and hence on the different stator
current levels.
Table 3.1 summarises the input conditions and the calculated
parameters. The parameters R-. and X1 are referred to the 6-pole
stator side whereas R„ and X„ are referred to the 2-pole stator
side. The parameters R and X (defined earlier) in this table are
both referred to the 6-pole stator side using the effective turns
ratio calculated in the previous test (Section 3.2.4.1).
The rated current levels of the 2-pole and 6-pole stator
windings are 1.4A and 4.OA respectively. Due to the high series
impedance of the experimental machine it was not possible to obtain
rated current levels on the two stator windings at low applied
voltages although the machine exhibits short circuit conditions.
93
This is evident from the observations of Test 3. It is seen that,
except in the test carried out at relatively high voltage levels
(Test 3), the parameters giving the total resistance and the total
leakage reactance (Tests 1 and 2) are nearly equal. It is
reasonable to assume that, as mentioned previously, Test 3 is
subjected to errors due to the use of higher supply voltage levels
on the two stator windings. It can be further observed that the
ratio of the 6-pole stator current to 2-pole stator current is
nearly equal to the experimental value of the effective turns ratio
calculated in the previous test. These test results show that
confidence can be placed on the various assumptions made and on the
consistency of the series parameters calculated.
3.2.4.3 Experimental and theoretical equivalent circuits
As was mentioned in Section 3.1, a difficulty arises when the
total leakage reactance X has to be divided among the three sections
of the equivalent circuit. It was initially thought that the
reactances, X. and X„, which are directly calculated from the tests,
be divided into three sections according to the distribution
observed in the theoretically calculated equivalent circuit shown in
Figure 3.10. The resistances R. and R„ can be simply separated into
the corresponding stator and rotor components using the measured
values of the stator winding resistances as it was stated before.
However, the effect of stray losses on the effective series
resistances can be best accounted for by distributing its effect in
all three sections of the equivalent circuit. This can be
accomplished by distributing the total measured series resistance
94
(R) according to the distribution of the resistances of the three
sections of the theoretical equivalent circuit (Figure 3.10). The
series parameters shown in Figure 3.11 have been derived from the
results of Test 1 given in Table 3.1.
2.2
6-pole stator Input
J2.77
1498
2.91 J8.31 -I | r r w v
J52
Jl.94 2.05
J95 620
FIGURE 3.10 THEORETICAL EQUIVALENT CIRCUIT AT STANDSTILL
All parameters in 12 and referred to the 6-pole
stator side
2.41 J2.79
6-pole stator Input 661
3.18 J8.36
J56.8
Jl.95 2.24
J103 1204
FIGURE 3.11 EXPERIMENTAL EQUIVALENT CIRCUIT AT STANDSTILL
All parameters in ft and referred to the 6-pole
stator side
95
The calculation of the parameters of the theoretical equivalent
33 34 circuit is based on well established methods ' . The airgap space
harmonics due to the rotor winding other than the 2-pole and the
6-pole waves were considered to act as leakage within the rotor.
The deep bar effect in the rotor bar sections embedded in the slots
was found to be insignificant in the calculations. The saturation
factors corresponding to the 2-pole and 6-pole magnetising
reactances were the same as those used in Chapter 2
(Section 2.6.3*1) in the calculation of coupling impedances. The
resistances representing the iron losses have been calculated using
the expressions developed and described in Chapter 4. These
expressions permit separate calculation of the stator and rotor iron
losses corresponding to either the 2-pole or 6-pole airgap flux
density waves in the airgap, at standstill. The two airgap flux
density levels chosen for this calculation correspond to the two
stator supply voltages obtained in the test performed to determine
the shunt parameters.
In comparing the theoretical and the experimental equivalent
circuits it can be noted that the total leakage reactance of the
former case is 13.0212 , whereas the total leakage reactance in the
latter case is 13.1012. The total resistance in the former case is
7.16ft and in the latter case it is 7.83ft. This difference in
resistances can be primarily attributed to the presence of stray
losses which simply cannot be accounted for in theoretical
calculations. There is also a close agreement between the
theoretically calculated and experimentally determined magnetising
ta
96
+» U o
• o-<D
— ID O
_ +»
a> L. o a> -C •**
\ -o Q) +> 10
— 3 U
— a o
u u
•
cr <D
10 •p
a <D
E •— c <D CL X <D
1
-a o +>
a —. 3 U
— a o
-o a> c 3 w 10
a> £
CO
OH 1—1
J CO
33 H l-H S= H z w 03 OI S U
o. o z o h-1
H < h-1
03 < >
N 33 O LO
+-> aj
. . Z
•J • —
> o LO
+J n3
TJ 0) •H r-i P. OH 3 w u o +J a3 +-= cn CD r-t
o p. CO
VS <U
• H
3 a M
• H
O 4J S-l
o X tn
u o +J aj +J cn OJ rH O OH I
CM
l-H
C5
W
a M Ex.
s •
""t
LO a
en
E3 •
CO
m •
CM
ca a
0J
(b) iN3yyn3
ta
97
+»
o o
I
a-(D
_ 10
o •— +> Q> L.
o a JC
•
1 -o ID +> 10
_ 3 O
ta— 10 O
•»
u o
• tr 0)
a *» c <D
e _ L.
o Q. X
<0 i
1
-a ID • * »
a — 3 U
ta— 10
o 1
1
"O <D
u 3 M 10 <D
£
<
OJ
CO
LO
CD
cn
CO
cn
ca
O, l-H
.J CO 33 H
OJ
Of 03
o H OH
O Z
o I — I
< h-1 03 > CO rH
eo
OJ
B o l-l OH
<s ca
c«N) anoyoi
98
reactances. It can be noted that although the calculated and
experimentally determined total iron losses agree reasonably well,
the resistances representing the iron losses due to individual flux
density waves do not agree between the experimental and the
theoretical cases. This error can be attributed to the
imperfections in the experimental techniques and set-up and
graphical interpolations used in the determination of shunt
parameters. Although this is the case, it can be considered that
for the overall equivalent circuit there is a good agreement between
the experimentally determined and the theoretically calculated
values of the parameters.
3.2.4.4 Performance characteristics
The accuracy of the parameters obtained experimentally can be
tested by comparing the performance predicted by the experimental
equivalent circuit with those obtained from a load test. It was
decided that the SFCIM be operated as an asynchronous motor supplied
on the 6-pole stator side with the 2-pole stator side short
circuited. Figure 3.12 illustrates graphically the variation of the
2-pole and 6-pole stator currents with slip. Figure 3.13
illustrates the variation of the torque. These figures represent
three different cases which are described below.
(a) measured;
(b) predicted using the theoretical equivalent
circuit of Figure 3.10; and
(c) predicted using the experimental equivalent
circuit of Figure 3.11.
99
Measured and calculated results using the theoretical
equivalent circuit were also presented in Figures 2.12, 2.13 and
2.14 of Chapter 2. The calculated results using the coupled circuit
model are not included in Figures 3.12 and 3.13 as they have been
compared previously (Section 2.6.4.2 of Chapter 2) with the
calculated results using the theoretical equivalent circuit.
Figures 3.12 and 3.13 indicate that there is a close agreement
between the theoretically calculated results and the results
obtained from the experimental equivalent circuit. In Figure 3.13
the measured torque-slip curve indicates that the machine possesses
a slightly larger leakage reactance than suggested by the calculated
value. As a result of this difference the slip at which the maximum
torque occurs is smaller than the corresponding theoretically
predicted value.
It was noted that the equal distribution of X.. and X' (X„)
between the corresponding stator and rotor sections of the
experimental equivalent circuit does not lead to results with
significant differences when compared with the curves given by
case (c) in Figures 3.12 and 3.13. It was thought, however, that a
further investigation is required to draw any conclusions. The
results of this investigation are discussed in Section 3.3.
3.3 Effects of the Redistribution of Leakage Reactance on
Performance
The experimentally obtained values of the leakage reactances X.
and X' can be separated, in effect, into three sections of the Li
equivalent circuit in any desired ratio. This is inevitable if the
100
design details are unknown. As noted earlier, in the estimation of
the series parameters of the equivalent circuit, the position of the
fictitious short circuit governs the ratio between X1 and X'. 1 Ci
Therefore, some doubt is raised concerning the separation of the
total leakage reactance (X = X^X') into the three sections of the
equivalent circuit. The variation of the following quantities with
a variable distribution of the leakage reactances was considered to
give useful results. These quantities include the starting torque,
maximum torque and the slip (s) at which the maximum torque occurs.
The variability of the distribution of the total leakage reactance
can be accomplished in one possible way, by following the definition
of the two ratios C , and C n xl x2
x x' C = 52 or (1 - C ) = -i (3.14) (x + x' ) Xi (x + x' )
sp sq sp sq and
(x + x' ) x' Cx2 = —^ ~ or (1 _ Cx2} = — (3,15)
A A
For a given total leakage reactance X, the ratio C 1 is varied A 1
between zero and unity amounting to a shift of the total stator
leakage reactance from 2p-pole section to 2q-pole section of the
equivalent circuit. The variation of the ratio C „ between zero and X Li
unity amounts to a distribution of the total leakage reactances X
between the stator and the rotor. A magnitude of unity for C „ X Ci
amounts to zero rotor leakage reactance and hence the total leakage
3.0
UJ
• CC
P 2.0
X ac
1.0
DC
cr i —
cn
m a x i m u m t o r q u e
s t a r t i n g t o r q u e
c „ 2 - a . 5
J I I I I I
101
.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0
Cxl
FIGURE 3.14.1 VARIATION OF STARTING TORQUE AND MAXIMUM TORQUE
WITH C . xl
l_J ID • tr o i—
HtaZ
ID
s: i — i
X
y
h-CC
D_ l-H
_J CD
1.0
• 9
.8
.7
.6
.5
.4
.3
.2
.1
.0 J L .0 -1
Cx2=0.5
-L 2 .3 .4 .5 .6 -7 -8 .9 1.0
Cxl
FIGURE 3.14.2 VARIATION OF SLIP AT MAXIMUM TORQUE WITH Cxl
3.0 r-
UJ CD C3 tr <=> 2.0
3 21
cr
u
cc cr h-
cn
1.0
maximum torque
starting torque
Cx l=0.5
J L .0 .1 .2 .3 .4 .5 .6 .7 .8 -9 1.0
Cx2
FIGURE 3.15.1 VARIATION OF STARTING TORQUE AND MAXIMUM TORQUE
WITH C _ x2
102
LU CD a ac CO i—
2: CD 21 t—i
X
cr 2:
i—
cr CL. 1—1
_i
cn
1.0
.9
.8
.7
.6
.<=,
.4
.3
• 2
J L
Cxi=0.5
J L . 1 .2 .3 -4 .5 .6 .7 .8 .9 1.0
Cx2
FIGURE 3.15.2 VARIATION OF SLIP AT MAXIMUM TORQUE WITH C x 2
103
reactance is allocated only to the sections of the equivalent
circuit representing the two stator windings.
Figure 3.14.1 shows the variation of the starting torque and
the maximum torque with a variable C 1 where the ratio C „ is X 1 X^j
maintained constant at 0.5. Figure 3.14.2 illustrates the
corresponding variation of the slip (s) at which the maximum torque
occurs. The variation of the same characteristics with a variable
C „ while C , is held constant at 0.5, is illustrated in x2 xl
Figures 3.15.1 and 3.15.2.
It is thus evident that the key characteristics considered are
only marginally affected when the total leakage reactance is
redistributed. This situation is very similar to that of the
conventional induction machine.
3.4 Summary and Conclusions
The experimental techniques described in this chapter can be
applied to SFCIMs having phase-wound rotors. The performance
characteristics predicted using the experimental equivalent circuit
agree reasonably with those obtained from the load test. It is
known that the equivalent circuit parameters are subjected to
variations depending on the operating conditions. Many factors
contribute towards such variations. A set of equivalent circuit
parameters estimated under a set of conditions or a specific
condition cannot be expected to represent the machine under all
operating conditions.
In using the experimental techniques, especially in the
104
estimation of the shunt parameters of the equivalent circuit, utmost
care is required to ensure that the errors due to voltage
fluctuations on the supply system do not affect the test results.
The estimated leakage reactances X. and X' need not be divided
into the three sections of the equivalent circuit, according to a
specific ratio unless the design details are known. A more
important aspect is that the total leakage reactance must be
accurately determined.
In the case of a conventional induction machine, iron losses of
the rotor are not generally represented in the equivalent circuit,
as the frequency of rotor flux pulsations is low near the
synchronous speed. However, in the case of the SFCIM, the frequency
of flux pulsations of the rotor is relatively high even when the
machine operates near cascade synchronous speed. It was thought,
therefore, that an independent study is required to examine the
variation of the iron losses in a given range of operating speeds.
The details of this study are presented in Chapter 4.
105
CHAPTER 4: SOME ASPECTS OF IRON LOSSES IN SFCIM
4.1 Introduction
The different components of iron losses in conventional
35 induction machines are well understood. They include
(a) losses in the yoke and teeth;
(b) edge-burr losses;
(c) end-region losses;
(d) flux pulsation losses in the teeth; and
(e) losses due to manufacturing imperfections.
The more refined theories and methods of calculation of each type
listed above, other than that due to manufacturing imperfections,
have now reduced the dependence on empirical factors which were
earlier required to reach an agreement between theory and practice.
Research which has been undertaken in the past in
identification and measurement of different components of iron or no
losses with regard to conventional induction machines ' indicates
that a similar study on the SFCIM needs a treatment in its own
right. The present study is therefore focused only on the iron
losses caused by the fundamental flux in the teeth and the yokes of
the SFCIM, operating specifically between zero speed and the cascade
synchronous speed. It also enables approximate calculation of the
resistances, representing the iron losses, in the equivalent circuit
given in Figure 3.1 of Chapter 3.
The fundamental frequencies of flux pulsations in the stator
and rotor of the SFCIM are different from those of a conventional
106
induction machine. In this chapter, a basis for a theoretical
evaluation of eddy current losses is established, which allows
calculation over a given speed range, and difficulties associated
with the development of a theoretical basis for the calculation of
hysteresis losses are discussed. If the airgap is relatively short,
the losses which take place in the teeth due to high frequency flux
pulsations caused by the slotting could be substantial. However,
this component is disregarded in the present theoretical work.
Experimental work was carried out on the (2+6)-pole SFCIM which
was described earlier. These tests justify the assumptions used in
the theoretical modelling and enable indirect evaluation of the iron
losses of the SFCIM at a given speed. The latter is based on the
assumption that the superposition of losses due to each of the two
flux density waves is permissible. However, no technique was found
for the direct measurement of iron losses of the SFCIM under running
conditions. As stated before, accurate measurement of iron losses
of the SFCIM calls for an independent study.
4.2 Frequencies of Flux Pulsations in Stator and Rotor
Disregarding the effects due to slotting and unwanted space
harmonics, the primary frequencies of flux pulsations present in the
yoke and teeth of the stator and rotor are determined by the 2p-pole
stator supply frequency w , rotor frequency w [Equation (1.2) of
Chapter 1] and 2q-pole stator frequency w [Equation (1.4) of
Chapter 1]. The variation of the frequencies wf and w with slip s
is illustrated graphically in Figure 4.1.
107
/N
f P
f P
frequency
FIGURE 4.1 FREQUENCY VARIATIONS IN THE SFCIM
-. = supply-frequency on 2p-pole stator = w /2ff P P ' = 2q-pole stator frequency = w /227
' - rotor frequency = w /2n
From Figure 4.1, it can be observed that the rotor experiences
a frequency of qf /(p+q) at the cascade synchronous speed of the
SFCIM and hence it will be subjected to a lower frequency of flux
pulsation if p > q, for a machine of given synchronous speed. The
stator yoke and teeth also are subjected to the two principal
frequencies of flux pulsation of f and f . P q
4.3 Flux Density Waves in the Airgap and Yoke
Assuming that the superposition of flux density waves is
permissible, the flux density [B (0,t)] in the airgap can be 8>
108
expressed in the form
B (6,t) = B sin(pe + a + w t) + B sin(qe - w t) (4.1) a p P q q x '
where 0 = spatial angle measured with respect to a point on
the stator frame
a = an initial angle
B = peak value of the 2p-pole airgap flux density wave
B = peak value of the 2q-pole airgap flux density wave.
The stator/rotor yoke flux density [B (0,t)] can be expressed in the J
form
B(e,t) = — \ BQ (e,t) de y 2y J
a
D
2y
B B cos(p6 + a + w t) + —— cos(q0 - w t)
p q P q (4.2)
where D = airgap diameter
y = yoke depth of stator or rotor.
4.4 Eddy Current Losses in Stator and Rotor
4.4.1 The analytical expressions
Prior to the derivation of expressions for the eddy current
losses in the stator and rotor cores, it is necessary to consider
two distinct operating conditions which arise. The stator yoke and
109
teeth are subjected, in one case, to two equal frequencies (f = f )
of flux pulsation at standstill and, in the other case, to two
unequal frequencies (f * f ) of flux pulsation when the machine is
running, due to the two flux density waves. The rotor is always
subjected to flux pulsations of the same frequency (f ) due to the
two flux density waves.
When f = f , the resultant flux density in the stator teeth P q [B.(0,t)] can be expressed in the form
Bt(0,t) = kt B + B - 2B B cos[(p+q)9+a] P q P q
1/2 sin(w t+£)
p
kt BR Sin( V+ C ) (4.3)
where
B R
a constant of the stator teeth of the machine
2 2 11/2
B + B - 2 B B cos[(p+q)0+a] P q P q
phase angle of the resultant flux density wave,
It therefore follows that the eddy current losses in the stator
teeth around the whole periphery (P ) can be obtained by the
integral
2J7
et k f k e p tj«2
d0 (4.4)
where k = eddy current loss coefficient. e
110
Substitution of the expression for B in Equation (4.4) followed by
the integration gives
P . = 277k f2 k. ( B2 + B2 ) (4.5) et e p t p q ' v*-j;
Thus, the average eddy current loss in the teeth is independent
of 0, because of the circular symmetry of the core. A similar
expression can be derived to represent the eddy current losses in
the stator yoke (P ) ey
[ r B .. _ r B ->2T (4.6)
r r B_ ,2 P = 277k r k ey e p y
where k = a constant of the yoke. y
The expressions given by Equations (4.5) and (4.6) are also
applicable to the rotor but with a modification applied to the
frequency of flux pulsation.
In the case where the frequencies of flux pulsation in the
stator due to two flux density waves are not the same (f * f ), the P q
eddy current losses due to each flux density wave evaluated at the
respective frequency can be directly added up assuming no saturation
and other mutual effects such as temperature effects.
Following the principles outlined above, the general
expressions for the eddy current losses in the teeth and yoke of
both stator and rotor are given by
Ill
P 4 = m,. k f ets ts e p L t J
s
( B 2 + B 2 s2) P q
(4.7)
N A ,2 n . „2 SS S P = m k f eys ys e p
H=-H I 1 227y J L
s
r B , B
«• q •>
(4.8)
P . = m. k f etr tr e p
2lElf [Al2 ( B2 • B2 ) p+q J I * J P q
(4.9)
P , m k f2 f3l£i
eyr yr e p [p+q
2 , N A rs r 2zryrJ
r B
P *• P
B v (4.10)
*• q
where
ets
eys
etr
eyr
m ts
m ys
m tr
m yr
N ss
N rs
eddy current loss in the teeth of stator
eddy current loss in the yoke of stator
eddy current loss in the teeth of rotor
eddy current loss in the yoke of rotor
mass of iron forming the stator teeth
mass of iron forming the stator yoke
mass of iron forming the rotor teeth
mass of iron forming the rotor yoke
slot pitch of stator
tooth width of stator
slot pitch of rotor
tooth width of rotor
number of stator slots
number of rotor slots
yoke depth of stator
112
y = yoke depth of rotor.
It is interesting to examine the manner in which the eddy
current losses vary in a conventional 2(p+q)-pole induction machine
built into a similar frame as the SFCIM. This is described in the
expressions given by Equations (4.11) through to (4.14). The peak
flux density in the airgap of this machine is assumed to be equal to
<Br,+BJ' p q
r A
ets m. k f ts e p p q
(4.11)
0 r N A .2
P = m k r ' faf s
eys ys e p 27Ty
r B +B ,2 p q
p+q (4.12)
P . = m, k f s etr tr e p
r- A
(B +B ) p q
(4.13)
P = m k f2 s 2
eyr yr e p
, N A ,2 , B +B rs r ] [-JLJL]
J L n+n J
277yr • l p+q
(4.14)
In the derivation of the expressions for eddy current losses in
the yoke, it was assumed that the flux lines in the yoke were all
circumferential. In conventional induction machines, an analysis of
the magnetic field in the yoke reveals that there are radial
components of flux lines . As the pole pair number and the depth
of the yoke increase, additional losses will take place in the yoke
due to radial flux lines. In the SFCIM, with respect to the p-pole
113
pair flux density wave these additional losses can be taken into
account by multiplying (using V instead of 'p' with regard to the
q-pole pair flux density wave) the corresponding eddy current losses
due to circumferential flux lines in the yoke, by the factor (C ) . e
°e= MdrH^H <4-15) u a+1 J L a -1 J
where a = ratio of airgap diameter to back of core diameter. The
ratio 'a' will be less than unity for the stator and greater than
unity for the rotor.
With reference to the experimental machine which was described
in Section 2.6.2 of Chapter 2, the magnitudes of C for pole pair
numbers 1, 3 and 4 are given in Table 4.1. These coefficients were
applied individually with respect to each individual flux density
wave considered in the expressions derived for the eddy current
losses.
pole
pair number
1
3
4
C e
stator
1.01
1.14
1.25
rotor
1.03
1.38
1.65
TABLE 4.1 VARIATION OF C WITH POLE PAIR NUMBER e
8.0 i-
cn co o _J
LU tr DC ID Cd >-a a UJ
total - CONVENTIONAL MRCHINE
•7 .6 .5 .4 .3
SLIP
FIGURE 4.2 VARIATION OF EDDY CURRENT LOSSES WITH SLIP
P = 3, q = 1, B = 0.3T, B - 0.2T. P q '
effect of radial flux in the yokes excluded
114
total - CONVENTIONAL MRCHINE
Pelr (SFCIM)
0 .9 .8 .7 .6 .5 .4 .3 .2 .1 .0
SLIP
FIGURE 4.3 VARIATION OF EDDY CURRENT LOSSES WITH SLIP
p = 3, q = 1, B = 0.3T, B = 0.2T,
effect of radial flux in the yokes included
115
The magnitude of the eddy current loss coefficient k was e
estimated using the characteristics of the steel used in the
experimental machine. By subtracting the specified hysteresis loss
at a flux density of IT at 50Hz from the total iron loss under the
same operating conditions, the eddy current losses can be separated.
This method is not the most satisfactory way of evaluating k , as it
depends on the frequency and degree of saturation, however, the
coefficient so calculated was assumed to be constant in this study.
4.4.2 Theoretical results
Expressions given by Equations (4.7), (4.8), (4.9) and (4.10)
were evaluated for p = 3 and q = 1 in order to examine the case
where the 6-pole stator side is supplied at its rated voltage and
frequency. The supply-frequency is 50Hz and the rated values of B
and B are 0.3T and 0.2T respectively which correspond to the rated q voltages on the two stator windings. Variation of the eddy current
loss of the SFCIM, in the range 1 > s > 0, is illustrated
graphically in Figure 4.2. Component eddy current losses Petgi
P , P and P of the SFCIM are also shown. Variation of the eys etr eyr
total eddy current loss of an equivalent 8-pole conventional
induction machine built into a similar frame as that of the
(2+6)-pole SFCIM is also illustrated in the same figure. Figure 4.3
shows the same curves as in Figure 4.2 with the correction factor Cg
applied on the expressions giving the yoke losses. It is evident by
comparison of Figures 4.2 and 4.3, that the effect of radial flux
lines in the yoke of the stator and rotor on total eddy current loss
is only marginal in the present case.
(T) en o
Ld DC DC ZD U
o Ld
\
r J! -H—CAH3—g ta n cp ci in g -y—gdd|IZgL_|__g_ !
.9 8 6 .5 .4
SLIP 1
FIGURE 4.4 VARIATION OF EDDY CURRENT LOSSES WITH SLIP
p = 1, q = 3, B = 0.2T, B = 0.3T, * p q
effect of radial flux in the yokes excluded
116
8.0 r-
totnl - SFCIM
6 -5 -4 -3
SLIP
FIGURE 4.5 VARIATION OF EDDY CURRENT LOSSES WITH SLIP
p = 1, q = 3, B = 0.2T, Bq = 0.3T,
effect of radial flux in the yokes included
117
The case where the 2-pole stator side is supplied at rated
voltage and frequency is examined by setting p = 1 and q = 3.
Appropriate values for Bp and Bq are therefore 0.2T and 0.3T
respectively. Figure 4.4 illustrates the manner in which the eddy
current losses vary over the range of slip considered previously.
Figure 4.5 illustrates the same losses with the correction factor C e
applied to the expressions for the losses in the yokes. However,
the effect of this correction factor is seen to be only marginal.
By comparison of Figures 4.2 and 4.4, it is evident, that for
the case where the SFCIM's 2-pole stator is supplied at constant
frequency, the total eddy current loss at synchronous speed (s = 0)
is nearly 2.8 times that of the case where it is supplied on the
6-pole side. At half synchronous speed this ratio is approximately
1.7 and, as expected, is equal to unity at standstill. The total
rotor eddy current losses decrease at a faster rate as the machine
speed rises from standstill when the 6-pole stator side is supplied
at constant frequency. At synchronous speed, the total rotor eddy
current loss will be one-ninth of that which will take place when
the 2-pole stator is supplied. This will be true whatever the value
of the ratio of two flux density levels. The stator yoke also
exhibits a lower eddy current loss as the speed increases when the
6-pole stator side is supplied. Equation 4.8 suggests that, with
the assumed flux densities, the eddy current losses in the stator
yoke at synchronous speed will be one-fourth of that which take
place if the 2-pole stator side is supplied at constant frequency.
Although the magnitudes are less significant, the stator teeth
losses will always be comparatively low when the
118
stator side with lower flux density level is chosen to be supplied
at constant frequency. At standstill, the eddy current losses of
the conventional induction machine are higher than that of the
SFCIM. If the large pole pair stator side is selected to be
supplied at constant frequency and as the speed increases, the eddy
current losses of SFCIM tend to be smaller than that of the
conventional induction machine.
4.5 Hysteresis Losses in Stator and Rotor
4.5.1 The analytical expressions
At standstill, the stator and rotor teeth are subjected to
hysteresis loss cycles of the same frequency due to the two flux
density waves and, therefore, the total hysteresis losses in the
teeth around the whole periphery of the machine can be obtained by
the integral of the form
277
P. . = k. f k. I B* d0 (4.16) ht h p t J R
where P,. = total hysteresis losses in the teeth around the ht
periphery
k. = hysteresis loss coefficient h x = Steinmetz index.
By assuming a Steinmetz index of 2, this integration can be
simplified considerably. This has been the case in past
investigations on iron losses . Although the index 1.6 is
119
considered to give more accurate results, it is subject to
variations with level of flux density and the magnetic material.
Therefore the index 2 was considered accurate enough for the present
investigations.
At standstill, the hysteresis losses of stator and rotor are
given by the Expressions (4.17), (4.18), (4.19) and (4.20).
A .2 \. = m. k, f [—1] (B2 + B2) hts ts h p [_ j v p q'
(4.17)
P, = m k, f hys ys h p
N A ss s
277y
, B , P
11 P
2 , B ,
L q
(4.18)
P, . = m. k, f htr tr h p
A .2
tf (B2 + B2) p q
(4.19)
P, = m k, f hyr yr h p
r N A rs r
277y
B ,
u P
, B 2n (4.20)
The definitions of the new variables P. . , P. , P, x and P. hts hys htr hyr
giving the hysteresis losses of the teeth and the yoke of stator and
rotor follow from those of the eddy current losses.
Similar to the previously given coefficient C , which accounts
for the additional eddy current losses due to radial flux lines in
the yoke, the coefficient C, required to account for the additional
hysteresis losses caused by the radial component of flux in the yoke
v 36 is given by
120
• aP + a"P 4p
~ ^P + 7^P ^TTLna (4'21)
• a^ - a (aF - a y) J
The hysteresis loss coefficient k, was estimated using the specified
hysteresis loss used in Section 4.4.1.
As stated before, when the SFCIM runs at a finite speed the
stator core will be subjected to two different frequencies of flux
pulsation and the algebraic summation of the hysteresis losses
calculated independently due to the two flux density waves is
strictly incorrect. The B-H loops under such circumstances can
interact with each other giving rise to additional losses, but, a
mathematical treatment of the hysteresis losses under such
conditions is a complex exercise and is beyond the scope of this
work. Despite the difficulties associated with calculation of the
hysteresis losses of the stator core, the rotor hysteresis losses of
the SFCIM can still be estimated under running conditions as the
rotor is always subjected to only a single frequency of flux
pulsation.
At the cascade synchronous speed, the stator core of the SFCIM
experiences a flux pulsation of constant frequency superimposed on a
d.c. bias flux. When a magnetic material is subjected to an
oscillating flux and a d.c. bias flux, the hysteresis losses tend to
increase beyond what is given by the usual Steinmetz formula. These
37 increased losses have been investigated and formulated by Ball as
P, = f ( k. + k. . B^ ) BX (4.22) h p h hi dc m
f a-1 P L n + 1
where B, = d.c. bias flux density
121
m
khl
v
= peak a.c. flux density
= hysteresis loss coefficient applied to symmetrical
loops
= additional coefficient depending on the material
= exponent of the d.c. bias flux density (= 1.9;
given in Reference 37)
= Steinmetz index (= 1.6; as given in Reference 37).
as
Table 4.2 shows the results extracted from Reference 37 and
illustrates the effect of d.c. bias flux on the hysteresis losses of
a sample of medium silicon steel.
B, /B dc m 0
1.0
2.0
3.0
4.0
5.0
hu hs
1.00
1.06
1.20
1.47
1.77
2.45
TABLE 4.2 VARIATION OF HYSTERESIS LOSSES WITH D.C. BIAS FLUX
DENSITY
P. = hysteresis losses due to the unsymmetrical loop
P. = hysteresis losses due to the symmetrical loop
Following the reasoning given above, it is evident, that expressions
cannot be developed to examine the variation of hysteresis losses of
cn
cn
o
cn i — i
CO Ld DC LU I—
cn
14.0
12.6
11.2
9.8
8.4
7.0
5.6
4.2
2.8
1.4
'•-total - SFCIM
1
-Phy8(SFCIM) total - CONVENTIONAL MACHINE
-f 1 1 1
PhyrlSFCIM)
X-—
Phte (SFCIM)
Phlr (SFCIM) ft—a—B—a—a-^a
x —
total - SFCIM-^
Phus(SFCIM)
Phli (SFCIM) \
rr-r^r-T-^-^^-^-^-^-^-^~^-^-^-^-^, 0 8 6 .5 .4 .3
SLIP
• 2 • 1
FIGURE 4.6 VARIATION OF HYSTERESIS LOSSES WITH SLIP
P
effect of radial flux in the yokes excluded
3, q = 1, B = 0.3T, B = 0.2T, P q
co
co
o
co i — i
CO Ld DC Ld h-CO >-
15.0
13.5
12.0
10.5
9.0
7.5
6.0
•--total - SFCIM
3.0
1.5
0.
•+-V-.
P h y s (SFCIM)
Phyr(5FCIM) •5;jpr__„ «.
total - CONVENTIONAL MACHINE
H 1 I 1
-H— -*---K--X-
total - SFCIM-.*
PhU6 (SFCIM)
Pht8 (SFCIM) \
-x-. -X-. -X—
,Pht8lSFCIM)
Phtr(SFCIM) ~x~ 6
1.0 .6 .5 -4 • 1
SLIP
FIGURE 4.7 VARIATION OF HYSTERESIS LOSSES WITH SLIP
P
effect of radial flux in the yokes included
3, q = 1, B = 0.3T, B = 0.2T,
14.0
12.6
S 11.2
01 CO o -d CO I — I
CO Id DC Ld h-CO
9.8
8.4
-total - SFCIM
total - SFCIM
total - CONVENTIONAL MACHINE
t 1 1 1
Phy 8 (SFCIM)
,Ph l r (SFCIM) Pbl8(SFCIM)
i f — B — a — a — a — a ^a—o—a—a—a—B—a—a—a—a j I
1.0 .7 .6 .5 .4
SLIP
a — a — a — B — a — a - =? .3 . 1
FIGURE 4.8 VARIATION OF HYSTERESIS LOSSES WITH SLIP
p = 1, q = 3, B, 0.2T, B = 0.3T, q
effect of radial flux in the yokes excluded
123
co co o
co p — i
CO Ld CC Ld h-CO >-zc
15.0
13.5
12.0
10.5
9.0
7.5
6.0
4.5
3.0
1.5
-total - SFCIM
•P|,y8 (SFCIM)
- * — X — x — K -
o — P h u (SFCIM)
total - SFCIM.
.total - CONVENTIONAL MACHINE
Phy8(SFCIM)--.A
,Phy r (SFCIM)
Pht s (SFCIM)
Phtr (SFCIM) \
if—a—B—B—a—B—a—a—a—a—B—a ^ c p — a — a — a — a — a — a — a -- a — a — a — a — a — a — a -, : • I • I • I > I i T i I i I 1 1
0 8 .7 1 .0 .6 -5 .4 .3 .2
SLIP
FIGURE 4.9 VARIATION OF HYSTERESIS LOSSES WITH SLIP
p = 1, q = 3, B = 0.2T, B = 0.3T, p q
effect of radial flux in the yokes included
124
the SFCIM over a range of operating speeds other than at standstill.
The hysteresis losses of the (p+q)-pole pair conventional induction
machine, however, can be conveniently calculated by appropriately
modifying Equations (4.17) through to (4.20).
4.5.2 Theoretical results
Figure 4.6 illustrates the results of the case where the 6-pole
stator side of the SFCIM is supplied at its rated voltage and
frequency. The variation of rotor hysteresis losses in the range
1 > s > 0 is illustrated graphically in this figure. Hysteresis
losses of the stator teeth and yoke at s = 1.0 are indicated in the
same figure. At s = 0, ignoring the effect of d.c. bias flux, the
calculated hysteresis loss of the stator teeth and yoke are also
indicated. Figure 4.7 illustrates the curves corresponding to the
above case where the correction factor C, is applied and Figure 4.8
illustrates the case where the 2-pole stator side is supplied at its
rated voltage and frequency. Variation of the same losses,
calculated incorporating the constant C, is illustrated in
Figure 4.9. The effects of radial flux lines in the yokes of the
stator and rotor are seen to be only marginal.
It is obvious, that the rotor hysteresis losses decrease at a
faster rate if the 6-pole stator side is chosen to be supplied at
constant frequency. Ignoring the effect of d.c. bias flux at
synchronous speed, the hysteresis losses of the stator yoke is
one-fourth of that in the case where the 2-pole stator is supplied
at constant frequency. This ratio depends on the relative flux
densities and the pole pair numbers. The stator teeth loss in the
125
former case is 2.25 times that of the latter. This ratio depends
only on the relative flux densities. Although there is an increase
in the stator teeth losses when the 6-pole stator side is supplied,
the total stator hysteresis loss is dominated by the yoke losses.
Combining the observations on the hysteresis losses of the stator
and rotor, it can be concluded that there is a clear advantage in
supplying the large pole pair stator side at constant frequency.
4.6 Experimental Determination of Iron Losses
4.6.1 Introduction
Direct experimental determination of iron losses due to the
fundamental flux density wave of a conventional induction machine is
impossible without having to separate it out from a total quantity
which may contain copper losses, friction and windage losses and
other types of iron losses. Therefore it is necessary to understand
the different types of losses encountered in an iron loss
measurement and the manner in which they are supplied by the
electrical and mechanical sources. In relation to conventional
induction machines a comprehensive discussion on these aspects is
given by Alger and Eksergian . In their work, the small power and
torque levels involved in the measurements have required special
instrumentation.
An experimental technique which could be adopted to determine
the iron losses of the SFCIM at standstill was described in
Chapter 3. This involved the SFCIM having a phase-wound rotor with
all windings intact. In contrast, the measurements described in
this chapter have been made on the SFCIM having the rotor with open
30.0
25.0
20.0 -
15.0
10.0
5.0 -
—
—
—
0.0 1
50.0 1
80.0 1
110.0
P total (IS)
/ ye total (SS)
yzrs-pol e (15) /^2-pol e (SS)
.-^6-pol 9 (IS)
^^-X'6 - p o 1 e (S3)
i i 140.0 170
L 2-POLE STRTOR VOLTRGE (V)
10.0 20.0 30.0 40.0 50.0 60.0
6-POLE STRTOR VOLTRGE (V)
FIGURE 4.10 VARIATION OF IRON LOSS WITH APPLIED VOLTAGE/S AT
STANDSTILL
127
circuited windings (empty rotor). The practical application of
these methods of testing are not satisfactory as the rotor circuits
have to be open-circuited.
4.6.2 Iron losses under standstill conditions
Measurement of iron losses at standstill enables the
examination of the validity of the use of the principle of
superposition in the derivation of theoretical expressions, given in
previous sections, in relation to the SFCIM (Sections 4.4.1 and
4.5.1). The iron losses measured on each of the two stator sides
while individually supplied (designated by IS) at 50Hz, were added
to give a total value. Then, the two stator sides were
simultaneously supplied (designated by SS) at 50Hz and the iron
losses recorded on each of the two stator sides were added to give a
total value. These experimental results, observed at different
combinations of voltage levels on the two stator windings, are
illustrated graphically in Figure 4.10. In the same figure,
140V(L-N) and 50V(L-N) correspond to the rated airgap flux density
levels of 0.2T and 0.3T of the 2- and 6-pole fields respectively.
It can be clearly noted that the total iron loss recorded under
simultaneous presence of the two rotating fields is less than the
sum of those recorded when the two fields are individually present.
To some extent this discrepancy dispels the application of the
principle of superposition in the theoretical calculations. The
manufacturer's curves of steel characteristics giving the variation
of iron losses with flux density level illustrate that the linear
extrapolation is not possible and that the iron loss due to
128
DC CD I— cr f —
to UJ _)
o Q_ I
CD
-z. o
13.0
12.0
11-0J=_
10.0
9.0
8.0
7.0
6.0
5.0
4.0
3-0
2.0
1.0
0 .0 .1 .2 .3
Bdc ON 2-POLE STRTOR SIDE (Wb/m2)
FIGURE 4.11 VARIATION OF IRON LOSSES MEASURED ON 6-POLE STATOR
WITH B , ON 2-POLE STATOR SIDE dc
129
simultaneously present, in-phase flux density waves is less than the
sum of the losses calculated considering each of the two flux
density waves individually. This trend agrees with the observation
made with reference to Figure 4.10. However, the difference in the
two top curves of Figure 4.10 was considered to be small enough to
be neglected for approximate calculations.
The existence of a magnetic cross coupling between the two
stator windings due to leakage fluxes and the altered degree of
saturation in various parts of the machine when the two fields are
present simultaneously can also be considered as reasons for the
anomaly arising from Figure 4.10. Much intensive flux plotting is
required to examine the flux distribution and its effects under
simultaneous presence of the two flux density waves.
Another test was carried out at standstill to examine the
effect of the magnitude of d.c. bias flux on the total iron losses.
The 6-pole stator winding was supplied at rated frequency (50Hz) and
the appropriately connected phase windings of the 2-pole stator were
supplied with a controlled direct current such that a calculated
level of flux density is produced in the airgap. The 6-pole stator
side airgap flux density was maintained at 0.3T and the 2-pole
stator side flux density was varied by the controlled direct
current. Figure 4.11 shows the variation of total iron loss
recorded on the 6-pole stator side with increasing 2-pole flux
density level (B, ) produced by the direct current. At a flux
density of 0.2T on the 2-pole stator side, which is also the rated
value, the total iron loss recorded on the 6-pole stator side is no
different from that observed with zero flux density on the 2-pole
130
stator side. This test was also carried out by interchanging the
role of the two stator windings, from which similar results were
obtained. These observations lead to the conclusion that the effect
of d.c. bias flux density on the total iron losses will be
significant only when its level is relatively high compared with the
alternating flux density. This follows the trend indicated by the
results given in Table 4.2. Therefore, calculation of the
hysteresis losses of the stator iron ignoring the effect of d.c.
bias flux at synchronous speed is justified.
4.6.3 Iron losses under running conditions
In a conventional induction machine, the stator iron losses,
rotor hysteresis losses and eddy current losses at standstill can be
separated experimentally from each other by assuming that the flux
pulsation losses due to permeance variations and space harmonics are
absent. The theory behind this test is described by Alger and O f
Eksergian . The technique adopted involves the measurement of the
stator input power while the induction machine, having an empty
rotor, is driven externally at various speeds and does not involve
measurement of shaft torques. At standstill, the stator input power
excluding the stator copper losses would be equal to the sum of
stator and rotor iron losses. As the speed is increased from
standstill in the direction of the field produced by the stator, the
stator iron losses and the hysteresis torque remain constant; while
the torque due to rotor eddy current losses decreases to zero
linearly as the synchronous speed is approached. As the speed of
the rotor is increased above synchronous speed the sign of the
131
airgap power due to iron losses in the rotor changes. The breakdown
of the total stator input power in this test is further illustrated
in Figure 4.12.
stator Input power
A
B
C
copper and Iron loss of stator hysteresis loss of rotor at standstill eddy current loss of rotor at standstill
7N
2B
zero speed synchronous speed
- >
speed
FIGURE 4.12 VARIATION OF INPUT POWER TO STATOR WITH SPEED
The same technique was applied to the SFCIM with each stator winding
supplied individually at the corresponding rated voltage and
frequency. Results of this test performed on the 2-pole stator side
are illustrated graphically in Figure 4.13. The supply
voltage (L-N) was maintained at 140V (at 50Hz) which corresponds to
the rated flux density of 0.2T. Results of the same test performed
on the 6-pole stator side are illustrated in Figure 4.14. The
supply voltage (L-N) in this case was maintained at 50V (at 50Hz)
cc LU
o Q_
I— ID
z 1—1
DC CD I—
cr h-If)
cn to o _i
DC Ld Q_ Q_ O r i
DC O I— cr i —
cn CD
z i — i
a CD _J CJ X UJ
DC UJ
cn cn o
DC Id
I-CJ
DC
cr
cn
cr
cn CD
• CD _J CJ X LU
21.0
18.0
15.0
12.0
9.0
6-0
3.0
.0
132
_] L .0 4.0 8.0 12.0 16.0 20.0 24.0 28-0 32.0 36-0 40.0
SPEED (RPM X 10-2)
FIGURE 4.13 VARIATION OF STATOR INPUT POWER WITH SPEED
2-pole stator supplied at 140V (L-N)
6-pole stator open-circuited
12.0
.0 3.0 6.0 9.0 12-0
SPEED (RPM X IB-2)
15.0
FIGURE 4.14 VARIATION OF STATOR INPUT POWER WITH SPEED
6-pole stator supplied at 50V (L-N)
2-pole stator open-circuited
133
which corresponds to the rated flux density of 0.3T. From the
(a) stator iron losses;
(b) rotor eddy current losses at standstill; and the
(c) rotor hysteresis losses at standstill
obtained from each figure (summarised in Table 4.3) it is possible
to estimate the same component losses in the SFCIM due to each flux
density wave at a known frequency of flux pulsation.
Stator iron loss (W)
Rotor hysteresis loss
at standstill (W)
Rotor eddy current
loss at standstill (W)
Due to 2-pole flux
density wave
12.30
2.45
3.29
Due to 6-pole flux
density wave
6.96
1.44
2.69
TABLE 4.3 COMPONENT IRON LOSSES
Table 4.4 numerically illustrates the iron losses of the SFCIM at
standstill due to each of the two flux density waves, calculated
both using the results of Table 4.3 and the theoretical expressions
presented in Sections 4.4 and 4.5.
134
Stator iron loss due
to 2-pole wave (W)
Rotor iron loss due
to 2-pole wave (W)
Stator iron loss due
to 6-pole wave (W)
Rotor iron loss due
to 6-pole wave (W)
*
Experimental
12.30
5.74
6.96
4.13
Theoretical
9.48
5.16
4.24
1.83
TABLE 4.4 BREAKDOWN OF TOTAL IRON LOSS (AT STANDSTILL)
DUE TO 2- AND 6-POLE FLUX DENSITY WAVES
calculated using the experimental results of
Table 4.3
Assuming that the superposition of experimentally measured
losses due to each of the two flux density waves is permissible, the
total iron loss of the SFCIM, at any given speed, can be estimated
using the fundamental relationships which describe the variation of
eddy current and hysteresis losses with frequency of flux pulsation.
When the 2-pole stator side is supplied at rated voltage (140V) and
frequency (50Hz) and at cascade synchronous speed, there will be no
stator iron losses due to the 6-pole flux density; and the rotor
frequency of flux pulsation would be 37.5Hz. Similarly, when the
6-pole stator side is supplied at its rated voltage (50V) and
frequency (50Hz), the 2-pole flux density wave would not cause any
iron losses in the stator at cascade synchronous speed and the rotor
135
frequency of flux pulsation would be 12.5Hz. Obviously, at
standstill, the iron losses of the SFCIM do not depend on the stator
side which is selected to be supplied at constant frequency,
Table 4.5 gives the experimental and theoretically calculated iron
losses of the SFCIM at
(a) standstill;
(b) cascade synchronous speed when the 2-pole
stator is supplied at rated voltage and
frequency; and
(c) cascade synchronous speed when the 6-pole
stator is supplied at rated voltage and
frequency.
Standstill
At cascade synchronous
speed, 2-pole stator
supplied
At cascade synchronous
speed, 6-pole stator
supplied
*
Experimental
iron loss (W)
29.13
18.60
8.30
Theoretical
iron loss (W)
20.71
14.46
5.58
TABLE 4.5 IRON LOSSES OF THE SFCIM AT STANDSTILL AND
CASCADE SYNCHRONOUS SPEED
Calculated using the experimental results of
Table 4.3
136
From the numerical results presented in Table 4.5, it is evident
that there is a definite advantage in selecting the large pole pair
stator side to be supplied at constant frequency. These results
also show that despite the simplicity of the basis for theoretical
calculation of iron losses at standstill and cascade synchronous
speed, there is reasonable agreement between the theoretically
calculated losses and the experimentally determined losses. The
difference can be attributed to the additional components of losses
present in the experimentally observed quantities, whereas the
theoretical calculations were based on the presence of the two
principal flux density waves only. As stated before, the
experimental estimation of the total iron losses of the SFCIM was
based on the assumption that the superposition of iron losses due to
each of the two flux density waves is possible. This simplifying
assumption may not hold true in practice in the present case.
4.7 Suuary and Conclusions
The iron losses due to the two principal fields in the SFCIM
have been considered in a detailed yet simple manner in order to
investigate the way in which the losses vary from standstill up to
the cascade synchronous speed. Theoretical derivations were based
on rotating flux density waves and commonly used expressions for the
eddy current and hysteresis losses. Difficulties associated with
the development of expressions for hysteresis losses have been
stated. The theoretical studies clearly indicate that the iron
losses, due to fundamental flux, which take place near cascade
synchronous speed can be kept to a minimum by supplying the large
137
pole pair stator side at constant frequency and allowing the smaller
pole pair stator side to carry the slip-frequency. This is
primarily due to the relatively low frequency of flux pulsation in
the rotor iron and relatively low yoke flux density on the large
pole pair stator side which is excited at constant frequency.
Experimental measurements of iron losses have been carried out
using unconventional techniques which, however, are not very
suitable from a practical viewpoint as the rotor circuits have to be
open circuited. The experimental results support the conclusion
that supplying the larger pole pair stator at constant frequency
results in a smaller iron loss at fractional operating slips.
The theoretical expressions enable the calculation of iron
losses at standstill, due to each individual field. This
facilitates the estimation of the parameters of the theoretical
equivalent circuit given by Figure 3.1 of Chapter 3. These
calculated values are given in Figure 3.10 of the same chapter.
Obviously, these parameters representing the iron losses have to be
adjusted depending on the speed. The decrease of iron losses with
speed is usually counter-balanced by the increase of friction and
windage losses in practice. The magnitude of the total iron loss in
a practical machine is a small percentage of the power rating and
hence the error involved in the corresponding parameters is not
reflected in performance calculations involving the equivalent
circuits. Thus, it is reasonable to assume constant values for the
resistances representing the iron losses in the equivalent circuit
for the SFCIM.
138
CHAPTER 5: DESIGN ASPECTS OF PHASE-WOUND ROTOR WINDINGS FOR
SFCIM
5.1 Introduction
The wide ratio pole pair combination p = 1 and q = 3 of an
SFCIM gives the highest possible cascade synchronous speed for a
given supply frequency and therefore is of special interest. For
such a pole pair combination the number of rotor slots per phase
group generally available for a 'multicircuit single-layer bar
winding' is relatively large but fixed. However, overall
performance of the machine is adversely affected by the poor
coupling between the smaller pole pair field and the inner U-loops
7 of a phase group . The effectiveness of the winding cannot be
varied other than by omission of one or more of the inner U-loops in
a phase group, which leads to under-utilisation of the available
slot space. Harmonic analysis performed on the 4-phase multicircuit
single-layer bar winding which was considered in the theoretical and
experimental work in Chapter 2 further illustrates the poor
performance of the winding for the wide ratio pole pair combination
p = 1 and q = 3. Results of the above harmonic analysis are given
in Appendix 4, from which it can be observed that the magnitudes of
the unwanted harmonics are high, leading to a rotor having a high
leakage reactance. Thus, the multicircuit single-layer winding is
unsatisfactory for SFCIMs having wide ratio pole combinations.
However, for SFCIMs having close ratio pole combinations, such as 18
and 12, the multicircuit single-layer bar winding has been found to
phase 4
phase 3
139 phase 1
phase 2
5.1.1
phase 4
phase 3
phase 1
phase 2
5.1.2
FIGURE 5.1 PHASE-WOUND ROTOR WINDINGS SHOWING THEIR END
CONNECTIONS
140
give good performance .
In contrast to the multicircuit single-layer bar winding the
phase-wound windings offer greater flexibility in their design and
hence can be considered to be a suitable alternative for SFCIMs
having a wide ratio pole combination such as p = 1 and q = 3. To
justify the need for an examination of the design aspects of
phase-wound rotor windings, the following description is given. For
the specific case of p = 1 and q = 3, Figure 5.1 illustrates two
possible phase-wound winding arrangements, each placed on a 44-slot
rotor. For convenience only the end connections are shown. In
Figures 5.1.1 and 5.1.2, each of the 4-phase windings takes the
concentric form (e.g. Figure 2.4 of Chapter 2), but they can also be
arranged as progressive windings (e.g. Figure 1.2.1 of Chapter 1)
which are electrically and magnetically equivalent to the concentric
form. The magnitudes of the harmonic winding factors (K ) and the wnr
corresponding percentage space harmonic mmf content of these two
winding arrangements are given in Table 5.1. The percentage
magnitudes of the space harmonic mmf content for the case where the
SFCIM having the 4-phase multicircuit winding (Section 2.6.2 of
Chapter 2) is supplied on the 6-pole stator side and operated at
s = 0.05, are also given in Table 5.1 for the purpose of comparison.
Detailed results of the harmonic analysis performed on the above
4-phase multicircuit winding are given in Appendix 4.
141
harmonic pole pair number
1
3
5
7
9
11
13
15
17
19
21
23
Winding in Figure 5.1.
K wnr
0.4070
0.7923
0.4704
0.0285
0.0156
0.1767
0.1631
0.0243
0.0382
0.0922
0.0780
0.0797
1
%mmf
100
64.9
23.1
1.00
0.43
3.9
3.0
0.4
0.6
1.2
0.9
0.9
Winding 5.
K wnr
0.5816
0.7049
0.0286
0.0387
0.0873
0.1472
0.1197
0.0054
0.1493
0.0560
0.0556
0.0567
in Figure 1.2
%mmf
100
40.4
1.0
1.0
1.1
2.3
1.6
0.0
1.5
0.5
0.5
0.4
multicircuit winding s = 0.05
%mmf
100
71.7
34.6
10.1
6.0
4.0
1.3
1.9
0.7
0.9
0.6
0.6
TABLE 5.1 HARMONIC WINDING FACTORS AND PERCENTAGE HARMONIC CONTENT
OF 4-PHASE WINDINGS
A cursory examination of Table 5.1 indicates that the relatively low
JLl.
magnitude of the 5 harmonic winding factor makes the partially
overlapping double-layer winding shown in Figure 5.1.2 favourable
when compared with the single-layer arrangement shown in
Figure 5.1.1, although some penalty has to be paid for the
under-utilisation of the available slot space in the former winding
142
arrangement. By a system of trial and error it is possible to
obtain alternative winding arrangements which may be equally
suitable. In the design of phase-wound rotor windings a more
rational approach is therefore required.
Because of the greater flexibility available in the design of
phase-wound rotor windings, numerous factors can be considered which
may improve the performance of SFCIMs having such windings. Two
such factors are low magnetising volt-ampere requirement and the
highest possible torque of the machine for a given frame size, which
will be shown to be dependent on the p- and q-pole pair winding
factors of the rotor.
The performance of the phase-wound rotor winding can be
examined initially from the viewpoint of magnetic fields by
considering the rotor winding factors in the manner which is
described in this chapter. The lengthy circuit calculations can be
used subsequently to examine the overall performance. An optimum
design of the phase-wound rotor winding configuration generally can
be accomplished independently of the stator windings for the SFCIM.
The stator windings then can be designed to match the rotor winding.
In relation to an SFCIM with the pole combination p = 1 and q = 3,
results of the investigations on a number of these aspects are
presented later in this chapter. However, the studies can be
conveniently extended for SFCIMs having other pole combinations as
well.
143
5.2 Single- and/or Double-Layer Smoothly/Discretely Distributed
Phase-Wound Rotor Windings
The spatial arrangement of one phase of a smoothly distributed
(p+q)-phase, phase-wound rotor winding is shown in Figure 5.2.
FIGURE 5.2 SPATIAL ARRANGEMENT OF A SMOOTHLY DISTRIBUTED
(p+q)-PHASE WINDING
Measured in mechanical units, the average pitch of the coils of a
phase group is 20 and the spread of a phase band is a. The
mechanical angle which corresponds to the number of smoothly
distributed slots allocated per phase group is 2ff/(p+q). Thus, for
the complete rotor winding to be a single-layer arrangement, the
144
conditions to be satisfied are given by the inequalities
a 9 > (5.1)
a n 9 + <
2 (P+q) (5.2)
which are illustrated graphically in Figure 5.3
partial double-layer winding
single-layer winding
B 2
double-layer winding
Err p+q
n- TT <*
> a
FIGURE 5.3 GRAPHICAL REPRESENTATION OF INEQUALITIES (5.1) AND
(5.2)
145
Figure 5.3 further illustrates the choices available for 0 and a,
for the rotor winding to be single-layer, double-layer or a blend of
the two types giving a partially overlapping arrangement as seen in
Figure 5.1.2. If the inequality of Equation (5.1) is violated, two
coil sides carrying currents which are 180° out of phase would be
lying in the same slot, thus serving no real purpose. This region
can be considered to consist of unused slots. For 9 - n/(p+q) and
a = 277/(p+q), the complete rotor winding would be a double-layer
arrangement.
The spatial arrangement of one phase group of a discretely
distributed, (p+q)-phase, phase-wound winding is shown in
Figure 5.4. The inequalities defining a single-layer winding
arrangement in a discrete slot rotor are given by
CP > B (5.3)
N CP + B < — (5.4) (p + q)
where B = number of slots per phase band
CP = coil pitch measured in number of slot pitches
N = total number of slots on rotor. rs
B
T I 146
fc
CP -a
FIGURE 5.4 SPATIAL ARRANGEMENT OF A DISCRETELY DISTRIBUTED PHASE
WINDING
5.3 Maximum Possible Torque Output from a Given Frame Size from
Magnetic Field Viewpoint
5.3.1 Dependence on winding factors
The expressions for maximum possible magnitudes of the
individual components (T and T respectively) of torque due to
p- and q-pole pair fields of the SFCIM built into a given frame size
can be obtained in a rudimentary way by ignoring the space phase
angle between the flux density wave and the corresponding rotor mmf
wave. The sum of T and T is considered to give a resultant pg qg
torque (T ) which is the maximum possible magnitude of torque
available from the SFCIM. It is noted that the omission of the
space phase angle and the addition of T and T together to obtain
a resultant is an oversimplification of the problem. In order to
simplify any further statements, the components T , T and Tg will
147
be referred to as «torque' although they do not refer to the actual
torque available from the SFCIM. Thus
P Trw* = n D L B F (5 5) pg 2 p pr lo-oj
q T rf = n D L B F ,5 fi) qg 2 q qr <0,t)'
where L = core length
F = 2p-pole component of the rotor mmf
F = 2q-pole component of the rotor mmf.
Naturally, to obtain the maximum possible 'torque' from the
machine, the flux densities B , B and the mmfs F , F should be P q pr qr
maintained at the highest possible levels. When the SFCIM is
supplied on the p-pole pair stator side at a given voltage and
frequency, the flux density B is explicitly specified, whereas the
flux density B is an implicit function of the pole pair numbers p
and q, corresponding rotor winding factors, and a factor which
accounts for the flux leakage within the rotor [Equation (5.12)].
If the SFCIM is doubly-fed the situation still remains the same, in
which case the applied voltage and the frequency on the second
stator winding has to be selected so that it matches the flux
density B . If the current in a phase winding of the (p+q)-phase,
phase-wound rotor winding is I (rms), the p- and q-pole pair r
148
component mmfs are given by
42 N K r o
n p V = (p+q) r-^ir (5.7)
U N K F„„ = (p+q) LJSLl (5.8) qr „ r
27 q
where N = number of turns per phase in a phase winding of the
rotor
K = magnitude of the 2p-pole winding factor of the
rotor winding
K = magnitude of the 2q-pole winding factor of the wqr
rotor winding.
Considering a 2(p+q)-pole conventional induction machine built
into a similar frame as the SFCIM and having an airgap flux density
of (B +B ), the expression for the maximum possible 'torque' (T ) p q eg
is given by
(P+q) T = 77 D L (B +B ) F (5.9) eg 0 p q c
where F represents the rotor mmf which is given by
J 2 NK F =m ^1 (5.10) ° n (P+q) °
149
In the expression given by Equation (5.10)
m = number of phases of the rotor
N = number of turns per phase of the rotor
K = magnitude of the (p+q)-pole pair winding factor of
the rotor winding
I = current per phase of the rotor winding(rms).
Using Equations (5.5) and (5.6), the expression for maximum possible
'torque' of the SFCIM can be normalised using that obtainable from
the conventional induction machine as given by Equation (5.9). Thus
T + T T p B F + q B F pg qg _ g . P Pr q qr (5.11) T T (p+q) (B +B ) F
eg eg p q' c
For an SFCIM having a single rotor winding, as was stated
before, it has been shown that the two flux densities B and B are
related by '
B p K 1 _P = wqr (5.12) B q K k, q ^ wpr 1
where k (< 1.0) is a factor which accounts for the series voltage JL
drop in the rotor section of the equivalent circuit given in
Figure 3.1 of Chapter 3. Using the notations given in Chapter 3,
and assuming that the SFCIM is supplied on the 2p-pole stator side
and the series voltage drop is due only to the rotor leakage
150
reactance
kx * _ J ! ! _ (5.13) X' + x'
mq r
Substituting F , F and F of Equations (5.7), (5.8) and pr qr c '
(5.10) in Equation (5.11)
Tg
Tcg
(p+q) Nr ir
m N K I wr c
r p + kx q ,
p k q
r- * XK~ wpr wqr
(5.14)
In Equation (5.14), the rotor winding factor K of the conventional wr
machine can be considered to be nearly equal to unity. The
quantities outside the square brackets, other than K , can be
expressed as a ratio of two linear current densities by assuming the
rotor windings of the SFCIM and the conventional machine to be
either single-layer or double-layer. The linear current density A
of the rotor of the SFCIM and the same, A , of the conventional c
machine can be defined by
2 N (p+q) I A = E L (5.15) 77D
2 N m I A = — (5.16) 77 D
151
w a .3
o
f-
<W
as o H U < Ct,
z a IT! H
O
W) J*
CC
o Z o l-H H < l-H
os < >
LO
• LO
CtJ
g C5 M te
cs
n
cr «.
l-H
II
ft
152
Therefore, it follows that
Ts Tcg
A r A K c wr
p + k^ q ..
' P k q ' K + X K wpr wqr
(5.17)
From Equation (5.17) it is clear that the maximum possible value of
the normalised 'torque' for the SFCIM is obtained when the
magnitudes of the winding factors K and K are equal to unity wpr wqr J
provided the constants A and A are equal. The factors A , A and r e r e
K can be considered as scaling factors for the purpose of
examination of the variation of the ratio T /T with the rotor g eg
winding factors K and K . It is difficult to design single wpr wqr
rotor windings having near unity winding factors with respect to 2p-
and 2q-poles when p is significantly different from q. However, by
selecting the coil pitch and spread in a systematic manner, it is
possible to obtain winding factors which are towards the upper limit
of the range.
The three-dimensional plot shown in Figures 5.5 illustrates the
manner in which the ratio T /T varies when the two winding factors g eg
are varied from zero to unity. The grid lines shown divide each
axis representing the winding factors into ten equal divisions. In
evaluating the surface, the ratio k, was assumed to be equal to
unity. This factor can be given a realistic value only when the
pitch and the spread of the coils in a phase group are both known.
The evaluation of k.. in such cases will be considered in
Section 5.3.2.
153
From Figure 5.5 it can be noted, that the winding factor
corresponding to the smaller pole pair number can vary in a
substantial range without affecting the magnitude of the normalised
'torque' whereas it is more sensitive to the winding factor
corresponding to the larger pole pair number. This difference
clearly becomes less significant for close ratio pole combinations.
The effect of the factor k, on the 'torque' can become significant
if the rotor series impedance voltage drop is substantial. In
practice, the total leakage impedance of the rotor and the stator
windings would have a direct influence on the magnitudes of the
current densities A and A . Furthermore the definition of these r c
two current densities [Equations (5.15) and (5.16)] would be
violated if the windings are neither fully single-layer nor fully
double-layer.
5.3.2 Dependence on pitch and spread
In the design of phase-wound rotor windings, the spread and
pitch of the coils can be considered as the key independent
variables which can be manipulated in order to achieve a better
performance. Thus, the winding factors K , K and the factor k, ' wpr wqr 1
can be computed for a given combination of 9 and a, and the
expression for normalised 'torque' given by Equation (5.17) can be
evaluated as a function of coil pitch and spread provided the factor
A is introduced into the calculations in a suitable manner. r
For the smoothly distributed phase winding with an average coil
pitch of 20 and a spread of a, as shown in Figure 5.2, the rotor
winding factor with respect to a general n pole pair space
154
BC CJ E-i h-1 PH
O z < a < cq OS a, CO
BC H h-1
& M
o H ^ s .
W) H
Ct,
o z o h-1 H < I—i OS < >
IO
LO
tt)
g o M ta
W)
c • H TJ
a •r-i
S T) cu •p 3 -a • H
M •*->
cn • H
Xi
>> rH J3 -p O O
s ca
TJ
TJ 3 rH
o c •H
M
EM O
c o
• H
• M
cd • H t-cd > ^
CT3
II
cr ».
l-H
II
ft
155
harmonic is given by
sin K wnr
= sin ne
na
T na
T
5.18)
Thus, the calculation of the two winding factors K and K is wpr wqr
straightforward. Assuming that the total rotor leakage reactance is
only due to the unwanted rotor space harmonics, an expression for k
can be established using Equation (5.13). Therefore
[K ..,
-F a—72—r~~nr
-F *l H IT—72" wnr
(5.19!
n?t p,q
Invoking the definition of the factor A given by r
Equation (5.15), the number of turns of a phase winding can be
assumed to be proportional to the coil spread a. The factors A and
K (a 1.0) which are related to the conventional induction machine wr '
can be assumed to be only scaling factors.
For the pole combination p = 1 and q = 3, taking 9 and a as
independent variables, the surface described by Equation (5.17) is
illustrated by Figure 5.6. The angles 9 and a are both varied from
77/10(p+q) to 277/(p+q) radian in steps of 77/10(p+q) radian.
The study carried out on the smoothly distributed rotor winding
can be conveniently extended to a discretely distributed rotor
winding. The number of slots of the rotor selected for
investigation is 44, which is the same as the number of slots
ft,
a
156
** > CM
II
P-CJ
l-H
II
PQ
BC CJ H r—i PH
a z < Q < W OS PH CO
EC E-i h-H 3c
bo O
fH ~-v
be H
Ct,
o z o l-H H < h-1 OS < >
t-•
LO
H
S CJ M Cx.
bo C •H
TJ
c •H £
TJ CU +J 3 rQ •H
tl +J cn
•r-i
TJ
!>> rH CU P cu M
CJ cn •H TJ
TJ CU TJ 3 r-i
o c •H
1—
_*
CM O
C o
•r-i
4-> cd • H t-l cd >
^ CO
II
cr
.. rH
II
ft
157
available in each of the rotors used in the experimental machine.
Independent variables used in relation to the discrete slot case are
B and CP which are indicated in Figure 5,4. For the purpose of
evaluation of Equation (5.17) taking B and CP as independent
variables, the factor A can be assumed to be proportional to B and
the factors A and K (^1.0) can be assumed to be scaling factors (—- w JL
only. Factor k. can be simply evaluated using the expression given
by Equation (5.19), for any combination of B and CP.
For the pole combination p = 1 and q = 3, Figure 5.7
illustrates the surface described by Equation (5.17) with regard to
the discrete slot case. In Figure 5.7, coil pitch (CP) is varied
from 1 slot pitch to 24 and the spread (B) is varied from 1 to 12
slots. This approximately covers the range of 9 and a considered in
the smoothly distributed case illustrated by Figure 5.6.
5.3.3 Conclusions
The similarity between Figures 5.6 and 5.7 suggests, that for
rough initial work, the study of a smoothly distributed,
hypothetical winding provides adequate information. From these
figures it is evident that there are different choices available for
the coil pitch and the spread of a phase winding which could provide
relatively higher 'torque' levels. However, some of these choices
could make the winding unsuitable. The reasons for rejection of
certain choices include long end connections and under-utilisation
of the available slot space. Another important reason being the
high harmonic leakage reactance, which will be considered in detail
in Section 5.6.
158
Referring to Figure 5.6, it can be seen that in the vicinity of
9 = 90 , a value for a can be selected to optimise the 'torque'.
However, this would mean a rotor winding with long end connections.
At 9 =* 32 and a s 64 another maximum point exists which would
yield a partially overlapping double-layer winding. As a result, in
the overlapping areas between two adjacent phase windings, the
linear current density for a given current increases. The numerical
output data corresponding to Figure 5.7 giving the values of the
three variables (B, CP and T /T ) provide exact values of B and CP g eg
which would yield the maximum 'torque' with reference to the 44-slot
rotor. These values for B and CP were both found to be equal to 8,
and gives a winding in which the adjacent phase groups partially
overlap. The corresponding 2- and 6-pole rotor winding factors were
found to be equal to 0.5122 and 0.5761 respectively. The leakage
factor k1 with this choice is found to be equal to 0.9598 which is a
good indicator of the magnitude of the harmonic leakage reactance
compared with the q-pole pair magnetising reactance
[Equation (5.19)] .
For the 4-phase, 44-slot rotor the number of slots allocated
per phase group is equal to 11. Thus, in order to obtain a
single-layer winding with best slot utilisation, the values for B
and CP can be selected to be equal to 5 and 6 respectively, which
corresponds to approximately 30% reduction in the 'torque' when
compared with the maximum 'torque' obtainable for the case where
both B and CP were equal to 8. The single layer winding with B = 5
and CP = 6 gives 2- and 6-pole winding factors which are equal to
0.4070 and 0.7923 respectively and the corresponding magnitude of
159
the factor k^ is equal to 0.8679 which suggests that this choice for
B and CP leads to a winding arrangement with a relatively higher
harmonic leakage reactance compared with the best possible case
which was obtained when both B and CP were equal to 8. Figure 5.1.1
illustrates the winding with a coil spread of 5 slots and a pitch of
6 slots. The phase-wound rotor of the experimental machine which
was described in Chapter 2 also used the same winding arrangement.
In the winding shown in Figure 5.1.2 the values of B and CP are
6 and 9 respectively. This arrangement produces 10% less 'torque'
than the optimum possible. The corresponding value of k, is 0.9755
which is higher than that which was obtained in the case where the
highest 'torque' was predicted.
Although this investigation provides useful information on the
spread and the coil pitch of the rotor winding from an
electromagnetic field viewpoint, other factors such as the
magnetising volt-ampere requirements and harmonic leakage reactance
must also be considered. These aspects will be discussed in greater
depth in the following sections.
5.4 Magnetising Volt-Ampere Requirements
Assuming that the SFCIM is singly-fed on the 2p-pole stator
side at a supply frequency of f the magnetising volt-ampere
requirements, Q and Q , corresponding to the p- and q-pole pair
160
sections of the SFCIM are given by
n 2 7 Q = _ B D L g f 10 (5.20) P 4 P P
11 2 7 Q = - B D L g f 10 (5.21) q 4 q P
and that of the conventional induction machine (supplied at a
frequency of f ) under comparison is given by
77 9 7
Q = _ (B + B T D L g f 10 (5.22) c 4 P q P
Using Equations (5.20), (5.21) and (5.22) the normalised value of
the magnetising volt-ampere requirement of the SFCIM is given by
2 2 Q + Q B + B _P <1_ - P q (5.23) Q (B + B )Z
c P q
From Equation (5.23), it can be seen that the normalised ratio of
the magnetising volt-ampere requirement takes the minimum value of
0.5 when the magnitudes of the two flux densities Bp and Bq are
equal. Therefore, according to Equation (5.12), the two winding
factors must be selected in proportion to the respective pole pair
numbers (when k± = 1.0). For the pole pair combination p = 1 and
q = 3, the rotor winding factor with respect to p = 1 must be
one-third of that with respect to q = 3. Thus, the constraint for
161
CO OS
o E-i
CJ
< Ct,
CJ Z hH
O
z DC H
Of
G> ft
O a*
M
Ct,
o z o hH H < hH OS < >
00
IA
w B u M Ct,
CO
11
cr «
r-H
II
ft
162
minimum magnetising volt-ampere requirement contradicts the need for
unity magnitude rotor winding factors which were required to obtain
the optimum 'torque' (Section 5.3.1).
To illustrate graphically the dependence of the normalised
ratio of magnetising volt-ampere requirement on the two winding
factors, by substituting Equation (5.12) in Equation (5.23) yields
Q + Q p K + (q K k,) P q _ wqr' VH wpr l'
- rr (5.24) Q (p K + q K k,)'
c ,r wqr H wpr 1
Figure 5.8 is a plot of the inverse of the expression described by
Equation (5.24) for p = 1 and q = 3 with the factor k set equal to
unity.
It is generally considered advantageous to design a machine
which gives the highest torque or the power, but having a minimum
magnetising volt-ampere requirement. This can be considered using
the 'torque' to magnetising volt-ampere ratio.
5.5 Maxiaua Possible Torque per Magnetising Volt-Ampere
5.5.1 Dependence on winding factors
Using the expressions given hy Equations (5.17) and (5.24) the
normalised expression for the 'torque' per magnetising volt-ampere
is given by
Y ( V V . Ar,p+ki'1) J5PT:'."W [-p qkj
T /Q A K eg' c c wr K
wpr-"
T--7WT <5'25) + K
wqrJ
163
cy
Of ar bO
o w
CO
OS
o H
o < Ct,
o n z DC H
cy ft
& bO 11
o H rH bo
Ct,
o z o hH
< hH
OS < > C7> •
in
w
HH tb
CO
II
cr
l-H
II
ft
164
Figure 5.9 illustrates the surface described by Equation (5.25) for
the pole pair combination p = 1 and q = 3. As stated earlier, the
factor k. can be given a realistic value only when the spread and
the pitch of the coils are known. Therefore k, was assumed to be
equal to unity. For the purpose of plotting the expression given by
Equation (5.25) the constants A , A and K were assumed to be only r c wr
scaling factors. From this plot it can be observed that the rotor
winding factor with respect to p = 1 can vary in a substantial range
without affecting the normalised 'torque' to magnetising volt-ampere
ratio whereas the same ratio is more sensitive to the rotor winding
factor with respect to q = 3. This is similar to the observation
made in Figure 5.5 of Section 5.3.1 in relation to the optimisation
of the 'torque'. From Figure 5.9 it can also be seen that there is
a possibility of satisfying the equal flux density criteria
(Section 5.4) while optimising the 'torque' to magnetising
volt-ampere ratio.
5.5.2 Dependence on pitch and spread
In a fashion similar to the study carried out in Section 5.3.2,
the dependence of the 'torque' per magnetising volt-ampere on the
coil pitch and spread is examined in this section. The factor k^ is
accounted for by using Equation (5.19). The surface described by
Equation (5.25), plotted considering a smoothly distributed winding
(Figure 5.2), is shown by Figure 5.10 whereas the same equation,
when plotted considering a discretely distributed winding
(Figure 5.4), is given by Figure 5.11. The range of variations
considered for 9 and a of Figure 5.10 and B and CP of Figure 5.11 is
165
o CM •**< fcj fej
II II
0 CD
DC CJ H
a z < < W OS PH CO DC H
cy
cy cy bC O
bO
o z o hH
H < hH OS < >
LO
Ex]
M tt,
bO C • H TJ 3
TJ CU +J 3 rQ • H
!H •P
cn •H
T) r-.
si •M
o o s cn
TJ
cu TJ 3 rH O
c
CM
O
c o •H cd •H cd
> CO
II
cr
i—i
II
ft
166
CXI
II
PH
CJ
r-1
II
CB
DC CJ H hH P.
Q Z <
a < w OS OH CO
BC H hH SS
-—. cr
cy + ft
cy * bO
H
o cy ~^ bO O
E-i
Ct,
o z o hH H < hH OS < >
bo 3 •H
TJ C •H *
TJ CU +J 3 rQ •H
M -M
cn •H TJ
>> rH UJ +J V (H
o cn •H
TJ
TJ cu TJ 3 ,—( CJ c •rH
.*
CM 0
3 O
•r-i
+J cd •H
M cd >
CO
II
cr -.
l-H
II
ft
LO
w
o I—I t b
II
ft, CJ
167
the same as before (Section 5.3.2). It can be observed that these
figures display characteristics similar to those of Figures 5.6
and 5.7 respectively, which represent the normalised 'torque'.
5.5.3 Conclusions
From the numerical output corresponding to Figure 5.11 the
optimum value for the 'torque' per magnetising volt-ampere in the
discretely distributed case is obtained when both B and CP are equal
to 7. It was noted that the same ratio for the case where both B
and CP are equal to 8 is only 1% less. This combination B = 8 and
CP = 8, coincides with the result noted in Section 5.3.3 with
relevance to Figure 5.7 when considering the maximum 'torque' alone.
5.6 Haraonic Leakage Reactance of Phase-Wound Rotor Windings
5.6.1 Dependence on pitch and spread
In the design of single rotor windings for SFCIMs, a greater
emphasis has to be paid to the reduction of unwanted space harmonics
of the rotor winding. It is known that the current loading of a
machine and its series parameters are interactive. Therefore,
minimisation of the rotor harmonic leakage reactance (x , ) (also rh
known as the 'differential leakage reactance') would directly assist
in the maximisation of torque output of a machine.
The factor k, calculated in the previous sections is a measure
of the relative magnitude of the space harmonic leakage reactance
x , of the rotor. The magnitude of this factor primarily depends on rh the pitch and the spread of the coils in a phase winding, therefore
an independent investigation of the harmonic leakage reactance would
168
Q < W OS OH CO
as H hH E£
W
u z < H CJ < w OS w o < « < w rJ
a hH
z o s OS < DC
Ct, o z o hH
H < hH OS < >
DC CJ H hH PH
o z <
bo 3 •H TJ 3 •H
S -3 0) H->
3 hO •iH
M •M
cn •H TJ
>> r — 1
J3 -P O O s cn
CM l-H
LO
CU
a hH tb
169
PH
CJ
CXI
Q < W OS p-co BC H hH
S
w CJ z < H CJ < w OS Cd O < « < w hJ
o hH
Z o s QS < DC Cb
O
z o hH
H < hH
OS < >
BC CJ H hH PH
Q Z <
bO 3
•r-i
TJ 3 •H
5
TJ CU P 3 rO •H
r. -P cn • iH TJ
>> r-i
<V P CU W CJ cn •H
TJ
eo l-H
LO
Ctl
g hH Cb
CO
ft
II
CQ PH
CJ
170
assist in the selection of the spread and the pitch of the coils of
the phase winding.
With reference to a smoothly distributed winding, the harmonic
leakage reactance of the rotor would be proportional to
2 _ f K .,2 , \ wnr
n*p,q
and with respect to a discretely distributed winding it is
proportional to
»n K .2 wnr
n*p,q
The variation of these two expressions with the spread and
pitch of the coils of a phase winding is illustrated by Figures 5.12
and 5.13 with regard to the smoothly and discretely distributed
cases respectively.
5.6.2 Conclusions
Examination of the numerical output data corresponding to
Figure 5.13 reveals that the choices available for B and CP giving a
low harmonic leakage reactance are not always the same as the
selections which can be made in order to maximise the 'torque' or
the 'torque' per magnetising volt-ampere. It is generally
considered best if the total leakage reactance can be minimised in
order to obtain the maximum torque possible from a given frame size
171
of a machine. This would also assist in operating the machine at
the maximum possible level of the current loading.
Considering the single-layer arrangement shown in Figure 5.1.1
in which B = 5 and CP = 6 as the base case, a comparison of the
relative magnitudes of the harmonic leakage reactance and the
corresponding 'torque' and the 'torque' per magnetising volt-ampere
ratios, for selected combinations of B and CP is given in Table 5.2.
The corresponding magnitudes of the rotor winding factors K , K wpr wqr
and the leakage factor k1 are also given in the same table.
B
5
6
6
6
7
7
7
8
8
8
CP
6
7
8
9
7
8
9
8
9
10
K wpr
P = 1
0.4070
0.4651
0.5247
0.5816
0.4599
0.5189
0.5751
0.5122
0.5677
0.6204
K wqr
q = 3
0.7923
0.7503
0.7447
0.7049
0.6686
0.6635
0.6281
0.5761
0.5454
0.4897
harmonic leakage reactance
1.00
0.53
0.26
0.19
0.47
0.31
0.22
0.37
0.33
0.31
kl
0.8679
0.9417
0.9697
0.9755
0.9515
0.9673
0.9723
0.9598
0.9604
0.9535
'torque'
1.00
1.24
1.29
1.28
1.33
1.38
1.37
1.42
1.40
1.32
'torque' per magnetising volt-ampere
1.00
1.18
1.17
1.11
1.23
1.23
1.16
1.22
1.15
1.02
TABLE 5.2 COMPARISON OF PHASE-WOUND WINDINGS WITH DIFFERENT SPREAD
AND PITCH
172
Referring to Table 5.2, it can be seen that the numerical
values giving the 'torque' or the 'torque' per magnetising
volt-ampere corresponding to different selections of B and CP vary
in a narrow range whereas the numerical values giving the harmonic
leakage reactance vary in a wider range. The primary reason for
this behaviour is that the selections made for B and CP have a
greater influence on the harmonic leakage reactance than on the two
winding factors K^ and K^ which are the controlling factors in
the expressions giving the 'torque' or the 'torque' per magnetising
volt-ampere although the leakage factor kj has been taken into
account.
The winding shown in Figure 5.1.2 corresponds to B = 6 and
CP = 9. This combination offers the minimum value of the harmonic
leakage reactance which is nearly 80% less compared with the base
case. When the adjacent phase groups of a phase-wound rotor winding
partially overlap (e.g. Figure 5.1.2) or fully overlap, in order to
obtain a magnetically and electrically balanced rotor winding, the
coil sides of one phase band have to be transposed with the coil
sides of the adjacent phase winding. Thus, where the coil sides
overlap with those of the adjacent phase groups, one set of coil
sides of a phase group will occupy the top of the slot whereas the
other set of the same phase group will occupy the bottom side of the
slots as in conventional double-layer windings. In the 4-phase
rotor under consideration, the currents in every other phase group
are in anti-phase and hence the mutual slot-leakage flux linkages
between phase groups do not cause a net induced voltage in any of
the phase windings.
173
For the purpose of comparison, calculations were made on a
hypothetical 44-slot rotor having a partially overlapping double
layer winding described by B = 6 and CP = 9. The slots which carry
two coil sides of adjacent phase groups were assumed to be twice as
deep as those slots which carry a single coil side. The bar type
conductor used in the hypothetical rotor winding was similar to that
used in the experimental phase-wound rotor winding which was
described earlier (Figure 5.1.1) and represented by the base case.
In the calculations, the deep bar effect was neglected. The
calculation of the slot-leakage inductance of the hypothetical rotor
winding showed that it was nearly 2.5 times that of the rotor
winding arrangement of the base case. This increase was due to the
larger slot depth and the increased number of bar sections embedded
in the slots. In the base case, the ratio of harmonic leakage
inductance to slot-leakage inductance was roughly equal to 15,
whereas in the hypothetical winding it was roughly equal to 1.2,
despite the increase of the slot-leakage inductance. Hence, as
expected, the dominant nature of the space harmonic leakage is very
much minimised when the coil spread and the pitch are appropriately
selected. Besides the slot-leakage fluxes and the space harmonic
leakage fluxes, the other component of interest is the end-winding
leakage flux. The contribution due to this was considered to be
negligible. The resistance of the hypothetical rotor winding
referred to the rotor itself was found to be roughly 1.5 times that
of the base case.
The performance of the hypothetical rotor winding can be best
examined by its incorporation with appropriately designed stator
174
windings. In Section 5.7, it will be shown that the rotor winding
factors have a marked influence on the design of the two stator
windings.
5.7 Design of p- and q-Pole Pair Stator Windings and Evaluation of
Performance
5.7.1 Design of stator windings
In this section emphasis is given to the fundamental design
aspects of the two separate p- and q-pole pair stator windings for
the SFCIM. Alternative single stator windings suitable for SFCIMs
are described in Reference 7.
The two separate stator windings can be designed so that either
of them can be supplied at a specified voltage at the same
supply-frequency producing a specified airgap flux density. This
situation is best described in the relationships given by the
equations
977 B D L
f£ N K f J- (5.26) J2 PS WPS P p
9rr B D L V = --L N K f -3 (5.27) q j qs wqs p
175
where Npg = number of turns per phase in the 2p-pole stator
winding
Nqs = number of ^rns per phase in the 2q-pole stator
winding
Kwps = magnit"de of the 2p-pole winding factor of the
2p-pole stator winding
Kwqs = magnitude of the 2q-pole winding factor of the
2q-pole stator winding
f = frequency of supply.
Usually, the winding factors K and K can be considered to be
nearly equal to unity for conventional stator windings.
Equation (5.12), with k{ set equal to unity can be used to rearrange
Equations (5.26) and (5.27) in the form
V^ N K p ps wqr
(5.28) V N K q qs wpr
Considering the mmf components F and F of the p- and a-pole ps qs ^ *
pair stator sections of the SFCIM
,„ N K = 3 ^ _ps_j^s (5<29;
PS 77 p PS
,„ N K F = 3 ii- qS wqs I (5.30 qS 77 a qs
176
where Ipg = load component of current (rms) in the 2p-pole
stator
Iqs = load comPonent of current (rms)in the 2q-pole
stator.
The rotor mmf components are given by Equations (5.7) and
(5.8). Using the mmf balance
K N I wpr ps ps = (5.31) K N I wqr qs qs
If the conductors forming the coils are operated at the same maximum
current density
K N a wpr ps p ,„ —J— - — (5.32 K N a wqr qs q
where a = cross-sectional area of the conductors in the
2p-pole stator winding
a = cross-sectional area of the conductors in the q
2q-pole stator winding.
For conventional stator windings Equation (5.31) also can be stated
as
K T a -WL-- pc p (5.33)
K T a wqr qc q
177
where T = turns per coil of the 2p-pole stator winding
T = turns per coil of the 2q-pole stator winding.
With a knowledge of the limiting values of the maximum airgap
flux density (B +B ), the magnitude of the two rotor winding factors ir T.
(K , K ) and the effective area per slot in the stator. wpr wqr '
Equations (5.12) and (5.33) can be used to solve for B , B , the P q
products T a and T a . Using the flux densities B and B in pc p qc q p q
Equations (5.26) and (5.27) together with the specified supply
voltages of the two stator windings, the number of turns required on
each stator winding can be found. It is then possible to calculate
the conductor cross-sections a and a , P q
In practice, some allowance has to be made so that each stator
winding can carry the magnetising current components in addition to
the load components. If the SFCIM is to be supplied wholly from one
stator side, that stator winding has to carry the magnetising
current of both p- and q-pole pair sections.
5.7.2 Evaluation of performance
Following the guiding principles stated in the previous
section, calculations were made on new stator windings which are
compatible with the hypothetical rotor winding discussed in
Section 5.6.2. The net slot area available for the new stator
windings and the limiting value of the maximum airgap flux density
(B +B ) were considered to be the same as in the experimental P q
machine, which represents the base case.
The equivalent circuit shown in Figure 5.14 gives the
178
theoretically calculated values of the parameters of the
hypothetical machine. The resistances representing the core losses
have been ignored.
1.79 Jl.91
6-pole
Input
2.51
J41.6
J2.0 J3.5 2.63
J196
FIGURE 5.14 THEORETICAL EQUIVALENT (STANDSTILL) CIRCUIT OF THE
HYPOTHETICAL MACHINE
All parameters are given in fi
and referred to the 6-pole stator side
When this equivalent circuit is compared with the theoretically
calculated equivalent circuit of the base case given by Figure 3.10
of Chapter 3, it can be noted that there is a 43% decrease in the
total leakage reactance and a 3% decrease in the total series
resistance. The theoretically predicted variation of torque with
slip of the hypothetical machine is given in Figure 5.15 for the
case where the 6-pole stator winding is supplied at 50V while the
2-pole stator winding is short-circuited. This torque versus slip
curve indicates that the high series resistance to leakage reactance
ratio of the rotor and the 2-pole stator section are responsible for
the high slip at which the maximum torque occurs. For the purpose
179
ca
E o
in •
*
ta •
• *
in •
cn
ca •
cn
C\J
en
t_ 3J
E o L
LD
CD
J L _ l
oo
CJ)
ca
m ca in • • •
cu c\J *-«
ca in ca
0_ 1 — 1
—1 CO
a, hH
J OT 33 H i — i
S=
w :=> CB O
Cb
o z o hH H < hH QH
< >
N 33 O LO
+J cd
. Z
J -»^^-> o LO
p
cd
-3 CU
•r-i
r-i
ft ft 3 cn IH 0 • P
c« P cn 0)
r — 1
O ft 1 to
VS <u p •H
3 O CH
•rH
a P tH O .3 cn SH
o p cd p cn 0) r-H
o ft 1
CM
LO
W
g hH tb
FN) 3nouoi
180
of comparison, the theoretical torque versus slip curve (Figure 3.13
of Chapter 3) of the machine representing the base case is also
included in Figure 5.15. The maximum torque obtainable from the
SFCIM represented by the equivalent circuit of Figure 5.14 can be
found to be approximately 75% more than that of the base case.
However, the torque available at low slips is not significantly high
compared with that of the base case, which suggests that the
resistances of the rotor and the 2p-pole stator have to be
considerably reduced.
An artificial decrease of the referred 2-pole stator resistance
caused the speed, at which the maximum torque occurs, to approach
the cascade synchronous speed without a significant change in the
maximum torque obtainable. Thus, by the provision of increased
copper area to the 2-pole stator winding and the rotor winding, the
machine could be operated near synchronous speed with a high
efficiency. An increase in the copper area of the 2-pole stator
winding naturally causes a decrease in the copper area available for
the 6-pole stator winding.
5.8 Conelus ions
Some design aspects of phase-wound rotor windings which are
especially suitable for SFCIMs having a pole combination of p = 1
and q = 3 have been considered. Being a single rotor winding, it is
desirable to have the highest possible values for the p- and q-pole
pair winding factors. This, however, does not ensure the best
performance from the SFCIM as the harmonic leakage reactance of the
winding can be high. It is important, therefore to minimise the
magnitudes of the unwanted space harmonics while the p- and q-pole
181
pair winding factors are maximised. In the process, the type of
rotor winding obtained is generally a double-layer winding with
partial overlapping. This essentially demands increased slot area
and may also leave some slots unoccupied. In practice, special
rotor stampings having two different slot shapes can be used so that
the bottom overlapping rotor bars are not subjected to deep bar
effect. If wider slots are used in the overlapping areas it is
essential that the tooth tips are not heavily saturated.
It was shown that the design of the phase-wound rotor winding
can be carried out independently of the two stator windings. The
design of the two stator windings is then greatly influenced by the
p- and q-pole pair rotor winding factors.
Investigations in this chapter lead to the conclusion that it
is essential to have single stator windings (e.g. the
'multi-parallel-path winding' described in Reference 7) with
improved performance for better overall performance from an SFCIM.
182
CHAPTER 6: THE SFCIM AS A BRUSHLESS CONSTANT-FREQUENCY
VARIABLE-SPEED GENERATOR
6.1 Introduction
Constant-frequency variable-speed generation has received much
29 attention in the past, especially in the aero-space industry .
Possible schemes include: the mechanical conversion of the
variable-speed to a constant-speed and the use of a standard
synchronous generator; and secondly, direct generation using the
variable shaft, speed. The reliability of the former scheme which
employs hydraulics, has been found to be less than desired. The
direct generation of constant-frequency using the variable shaft
speed has received a renewed interest in recent years with the n c nn
advent of solid-state power conversion equipment ' . This scheme
is also ideally suitable for wind-energy generation.
When a wound rotor induction machine is excited from the
primary side and driven, the secondary side experiences a
variable-frequency. As the shaft speed changes, to maintain the
frequency of the secondary side constant, the primary excitation
frequency has to be constantly adjusted. In practice, both the
primary frequency and voltage have to be adjusted in order to
maintain the flux level constant. However, the presence of slip
rings in the wound rotor induction machine makes the scheme
commercially unsatisfactory.
The SFCIM can be doubly-fed and offers the possibility of
constant-frequency, brushless, variable-speed generation. The basic
183
frequency-speed relationships applicable to variable-speed
constant-frequency generation using the SFCIM are given by
Equations (1.1) through to (1.7) of Chapter 1. If it is assumed
that the 2p-pole stator side delivers the constant-frequency output
at w , using Equation (1.7) it possible to determine the manner in
which the q-pole pair stator supply frequency w should be varied as q
the shaft speed w changes. This scheme is described in relation to
an ordinary two machine cascade connected induction machine system
and its dynamic behaviour under closed-loop control is reported by
Ortmeyer and Borger in Reference 27. Their investigation assumes a
limited variation of shaft speed which is the case in aircraft
applications. However, none of the available literature reports the
steady-state characteristics of the system over a wide range of
speeds nor has considered the SFCIM in place of the two machine
cascade system. This chapter investigates the steady-state
characteristics of the SFCIM having a 'multicircuit single-layer bar
rotor winding', operating over a wide speed range as a
constant-frequency generator and the theoretical and experimental
results are discussed. It is shown, that the steady-state analysis
of the SFCIM, under the present operating conditions can be carried
out using the coupled circuit theory developed in Chapter 2.
6.2 Variation of Slip with Angular Speed
An SFCIM operating as a variable-speed constant-frequency
generator is shown in Figure 6.1. The p-pole pair stator is
connected to the passive load and the q-pole pair stator is supplied
by the variable-frequency variable-voltage supply (inverter).
184
load
> i i * ,2p-pole stator output
SFCIM
i i i i i
Inp '2q-
Inverter
prlne
mover
Jt -pole stator
V
FIGURE 6.1 SFCIM AS A CONSTANT-FREQUENCY VARIABLE-SPEED
GENERATOR
Depending on whether p > q or p < q, the variation of the slips
s and s with w can be represented in two different graphs as q m
shown in Figures 6.2 and 6.3. The range of angular speed variation
considered is from zero to 2w /(p+q) (i.e. from zero to twice the
cascade synchronous speed).
ws
u n
185
FIGURE 6.2 VARIATION OF s , s AND s WITH ANGULAR SPEED P q
FOR p < q
1.0
u m
FIGURE 6.3 VARIATION OF ap, sq AND s WITH ANGULAR SPEED
FOR p > q
186
For the case p > q (Figure 6.3) it can be seen that when the
angular speed of the machine is equal to w /p, the slip s is zero P P
This amounts to a flow of direct current in the rotor winding of the
SFCIM in order to obtain an output voltage at constant frequency
from the 2p-pole stator. In the case of a wound rotor induction
machine operating as a variable-speed constant-frequency generator
it is possible to supply the direct current to the rotor at its
synchronous speed so that the constant frequency output is delivered
by the stator. However, the same is not possible with the SFCIM.
Therefore, if the shaft speed of the SFCIM is expected to vary in
the range considered, in order to avoid the above situation, the
large pole pair stator side has to be supplied by the
variable-frequency variable-voltage excitation source and the small
pole pair stator side has to be used to obtain the
constant-frequency supply.
6.3 Real and Reactive Power Flows
The real and reactive power flows into and out of the SFCIM
working in the generator mode can be understood using the
relationships derived in Appendix 3 in relation to the conventional
wound rotor induction machine (WRIM).
Assuming the SFCIM to be lossless, it follows from
Equation (A3.6) of Appendix 3, that when 0 < s < 1.0
(sub-synchronous operation), the inverter has to supply the real
excitation power to the q-pole pair stator winding at a variable
frequency of sw , so that the p-pole pair stator winding can deliver
the power to the external load at a frequency of w . When s < 0
187
(super-synchronous operation), the inverter has to absorb real power
while the p-pole pair stator delivers power to the external load.
At s = 0, the situation is very similar to that of a conventional
synchronous generator. By keeping the value of the slip s small,
only a small fraction of the power delivered to the external load
has to be handled by the inverter. In windmill applications,
however, this is a difficult requirement to satisfy if generation
has to be carried out over a wide speed range.
Other than at s = 0, the lagging reactive power required by the
SFCIM and the load generally has to be supplied by the inverter
unless compensating elements such as static capacitors are connected
across the load. However, care is needed to avoid self-excitation
of the machine which can lead to unstable operation.
6.4 Generating Scheme for Windmill Applications
A generating scheme suitable for windmill applications is
represented in Figure 6.4. The scheme shown is ideally suitable for
small-scale isolated generation of electrical energy. One primary
requirement of such a scheme is that the extracted energy must be
stored in batteries for later use while supplying the load at a
constant-frequency and constant-voltage.
The inverter is pulse-width modulated using the speed signal
derived from the mill so that both the inverter output voltage and
frequency are controlled. Power generated by the SFCIM is partly
supplied to the electrical load connected to the three-phase busbars
and partly used to charge the batteries supplying the inverter. The
188
load
control signals
controlled rectifier
SFCIM uilndnlll
Inverter S
control signals
FIGURE 6.4 GENERATING SCHEME FOR WINDMILL APPLICATION
189
charging current required for the batteries is controlled by the
phase-controlled rectifier. At zero mill speed the power required
by the load on busbars is totally supplied by the batteries via the
inverter and the SFCIM simply functions as a static transformer.
This is considered to be the outstanding feature of this scheme
where a single inverter is able to perform a dual function of
supplying the load and the excitation power of the SFCIM. The
absence of slip rings in the SFCIM makes the scheme suitable for
small-scale generation where low maintenance is a key factor. In
this chapter attention is focused on the steady-state operating
characteristics of the SFCIM working as a generator, on open-loop.
6.5 Theoretical Analysis
The theoretical model developed in Chapter 2 can be simply
adapted for the analysis of the operation of the SFCIM as a
variable-speed constant-frequency generator. The required equations
for this purpose are (2.36) or (2.62) of Chapter 2, depending on the
type of rotor winding used in the SFCIM.
As the generator output is obtained at an angular frequency of
w which is constant, the passive load (ZT) connected to the p-pole P L
pair stator winding estimated at the same frequency is algebraically
added to the first element of the first row [R + jwp(Lp-Mp)] of
these equations and the voltage V is set equal to zero. The
voltage V is set equal to the excitation voltage provided by the q
inverter and the slip 's s ', which is equal to the composite slip P q
's\ is varied according to the speed at which the windmill is
driven. Thus, the form of equation required for the analysis of the
CX)
190 IM
u
S CT + P,
b CT
S3
CT + 0.
CM
3
p.
IM
3 o
3
+
3 O
rH IM
CM
3
CM 3
b
b P.
b
CT CT + P, 3 "-3
CT + P.
3 H
P.
JO rH CM
3 H
3 •r-3
+
CM
3 11
X + CM
3
,P
CT fl P.
b P.
CT + P.
b CT
CT + P.
.P
s + 3 H
"P.
3 H
b CT
b
CT
b P.
b P.
P.
1 II
1 1
CT a P,
191
SFCIM having a multicircuit single-layer bar winding is given by
Equation (6.1).
The current I , multiplied by Zf , can be used to estimate the P L
voltage across the load connected to the 2p-pole stator winding and
the current I can be used to estimate the real and reactive power
flows into or out of the inverter; thus enabling the calculation of
the operating characteristics of the generator. Multiplying the
second row of Equation (6.1) by s s and making it equal to zero,
the operation of the SFCIM at cascade synchronous speed [w /(p+q)]
can be analysed. In this case the q-pole pair stator has to be
supplied with direct current. The effect of compensating capacitors
can be included in the model by considering them as a part of the
passive load Z..
In the theoretical calculations, when the generator operates at
sub-synchronous speeds ( 0 < s < 1 ), the reactive power flowing
into the q-pole pair stator from the inverter carries a positive
sign - indicating a lagging situation - which agrees with the
convention generally adopted. When it operates at super-synchronous
speeds (s < 0 ), which requires a reversal of phase-sequence of the
supply from the inverter in order to maintain the output frequency
constant, the reactive power flowing into the q-pole pair stator
winding carries a negative sign, and should still be interpreted as
a lagging reactive power flowing into the 2q-pole stator. This
anomaly is discussed further in Appendix 3.
6.6 Theoretical and Experimental Results
The SFCIM having the multicircuit single-layer bar rotor
192
L — LU CC DC U
Q
•
IA) 30U110A QH01 w — (A) 30U110A QH01
cc DC
u •
cx
» LC • LC
C_)
r • O
cr Q > -1
J
•
a •
n
cr
H
w P3
a < o -J 33 H hH S=
w C3 < H J o > Q < O .J Cb
o z o hH H << hH OH
< >
io •
10
H g u
cc cc u •
cr •
tb
-* (M
(A) 30B110A OUOT (A) 30U110A 0U01
193
winding which was described previously was chosen for the
theoretical and experimental investigations. The 2-pole stator
winding was selected to deliver the constant-frequency output while
the 6-pole stator winding is supplied from the variable-frequency
variable-voltage supply. This choice eliminates the situation
described in Section 6.2 where the SFCIM's rotor has to carry direct
current.
In the experimental set-up, the 6-pole stator was supplied from
a three-phase synchronous generator driven by a variable-speed drive
and the SFCIM was driven by a separate variable-speed drive.
Depending on the speed of the latter drive, the frequency and the
output voltage of the synchronous generator were independently
adjusted. The resistive load connected to the 2-pole stator was
varied in steps while maintaining the speed of the SFCIM, and the
frequency and voltage of the synchronous generator constant at the
desired levels. The excitation current on the 6-pole stator was
restricted to roughly 4.OA and the load current on the 2-pole stator
to roughly 1.OA because of continuous excessive heating of the
machine.
Figure 6.5 depicts the theoretical and experimental variations
of load voltage with load current, for four different values of
slips. The magnitude of the 6-pole stator side excitation voltage
(V ) given in each case is the experimental value which results in
an open-circuit voltage of 140V. The theoretical curves were
obtained using the same excitation voltage. A close agreement can
be seen between the theoretical and experimental results. The
excitation voltage required to obtain a given open-circuit voltage
100.0 r-
90.0
80.0 h 194
FIGURE 6.6 VARIATION OF EXCITATION VOLTAGE REQUIRED TO OBTAIN AN
OPEN-CIRCUIT VOLTAGE OF 140V WITH SLIP
100.0 r-
eslcui aied
FIGURE 6.7 VARIATION OF VOLTAGE REGULATION WITH SLIP
195
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LU DC CC r> (_) • cr • _i
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2 LU DC DC
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F-Z UJ 03 « n n Q < o j
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(0) 1N3UUH3 N0IiblI3X3 (U) lN3UUfl3 N0I1U1I3X3
196
on the load side can be found to increase with increasing magnitudes
of slip. The theoretically calculated variation of the excitation
voltage with slip, required to obtain an open-circuit voltage of
140V on the load side, is illustrated graphically in Figure 6.6 for
both sub-synchronous and super-synchronous operation. This
behaviour can be readily inferred from Equation (A3.4) of Appendix 3
in relation to conventional wound rotor induction machines.
Figure 6.7 illustrates the theoretically calculated variation
of voltage regulation taking 1.OA as the full-load current on the
2-pole stator side. For each value of slip at which the voltage
regulation is computed, the excitation voltage on the 6-pole stator
side has been maintained at a value which gives an open-circuit
voltage of 140V. From Figure 6.7 it can be observed that at speeds
close to cascade synchronous speed, the voltage regulation tends to
be extremely poor. In practice this can be partly rectified by over
exciting the machine thus boosting the flux level on the 6-pole
stator side. However, care is needed to ensure that the machine is
not saturated by an increase of the flux level, which may also lead
to excessive heating of the machine. In the design of the machine,
provision must be made to over excite the machine. Naturally, the
minimised magnitudes of the resistances and leakage reactances would
assist in maintaining a good voltage regulation.
Figure 6.8 illustrates the variation of the excitation current
with load current, for the same cases considered above. Both the
experimental and theoretical results agree closely.
The curves in Figure 6.9 illustrate the variation of the
excitation power and the power delivered to the load with load
197
cc cc
•
cr o
B
le
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L
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* 198
(M) 3SBHd / SS03 U3dd03 fM) 3SBHd / 9S0H 03dd03
LU DC CC _3 LJ
LJ U_ CD _l
H Z W ca ca ^ f)
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K E-< hH * CO LU CO CO
o -J ca w PH
0-o u Cb
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LU DC LC ZD LJ O (X O
CC DC =3 U
o cr •
< hH
ca < >
LO
ED
s a hH tb
l\l «
(M) 3SbHd / S901 U3dd03 (M) 3Sblld / SS01 U3dd03
199
current. Close agreement between theoretical and experimental
results also can be observed. As the slip increases positively, the
span of load currents over which the load power is found to be in
excess of the excitation power tends to decrease, implying that the
copper losses of the machine become greater than the mechanical
power converted to electrical power. Examination of Figure 6.10,
which illustrates the variation of the theoretically calculated
copper losses of the machine for the same cases considered above,
reveals that the variation of the magnitude of the copper losses is
not significantly different between the separate cases. Thus, as
the slip increases positively, most of the electrical power required
by the load is supplied by the inverter. For negative slips, the
real excitation power is negative except near the light load region,
implying that the inverter receives it while the load is supplied
with real power. Thus, for negative slips, the machine is more
capable in converting mechanical power to electrical power.
In the experimental machine under consideration, the copper
losses of the machine are significant compared to the load power and
thus it exhibits a poor performance. The conversion efficiency of
the machine can be expressed as the ratio of output load power to
the sum of input mechanical power and input excitation power. The
theoretically calculated conversion efficiencies for the four cases
considered above are illustrated in Figure 6.11. Obviously, a
higher conversion efficiency can be achieved only if the copper
losses of the machine can be reduced.
The experimental and theoretical variation of lagging reactive
power drawn from the excitation source with varying load current is
200
LU DC CC CD t_)
o cr o
z LU DC DC CD
u a cr o
Z J13N3I3IJ33 N0ISy3AN03 7. A3N3I3IJ33 NOISdSANOD
H Z H ca ca CD U Q < O DC
U
z w hH CJ hH Cb Cb
cq
z o hH CO
ca w >
z o u
LU DC DC
n LJ CD CE CD
7. A3N3I3I3J3 N0IGy3AN03 7. JLDN3I3IJJ3 N0ISU3AN03
201
-I I I I I I I i I i L J L
LU DC
az zz> LJ • CE CD
(JBM 3SBHd/ y3MOd 3AI13H3U H ° M 3SbHd/ y3M0d 3 A U 3 B 3 U
L —
I ' i
B in
.J L i I I
LU CC LC CD LJ
CD CE CD
J I. I _J L
cc
LU CC CC Z2 u CD cr u _j
>-CQ
O
w hH
-J
a, Cb CD co
ca w ^ o Cb
w > hH
H CJ < cq ca a z hH
a a <c j
Cb
o z o hH
H <c hH
ca < >
F-Z cq
ca ca > o
o J DC H hH
3= cq O ca CD O CO
z o hH
< H hH C) XI cq
cq BC E-
B ra si
m M -tn rj —
CN
LO
LU tb1
cc pa cc 5 •=> a LJ hH
Cb CD
cr CD
(JQM 3SbHd/ y3M0d 3AI13B3y [JB*) 35UHd/ U3H0d 3AI13B3y
202
shown in Figure 6.12. The general trend is that the lagging
reactive power required by the machine increases with the absolute
magnitude of the slip, for a given load current. At cascade
synchronous speed the machine has to be excited with direct current
and, therefore, the situation is similar to that of a conventional
synchronous generator. Observations made with respect to the
reactive power flows of the SFCIM working as a variable-speed
constant-frequency generator agree with Equation (A3.7) of
Appendix 3.
6.7 Summary and Conclusions
The steady-state operating characteristics of the SFCIM working
as a variable-speed constant-frequency generator have been
presented. A theoretical model to represent operation in this mode
has been obtained using the coupled circuit model developed in
Chapter 2. Its use is extremely simple and the various quantities
calculated can be directly employed to obtain important operating
characteristics. The theoretical and experimental results have been
found to agree closely.
It can be seen from the expressions given by Equations (A3.3)
and (A3.5) respectively of Appendix 3, that the phase angle of the
open-circuit voltage and the short-circuit impedance of a
conventional wound rotor induction machine operating as a
variable-speed constant-frequency generator are dependent on the
slip at which the machine operates. This dependence can be
minimised by keeping the rotor winding resistance as small as
possible, according to the same equations. The above behaviour can
203
be equally extended to the case of the SFCIM operating as a
variable-speed constant-frequency generator. The minimisation of
all resistances would also assist in raising the conversion
efficiency.
The lagging reactive power requirements of the machine and that
of the load can be supplied from external compensating elements such
as static capacitors or solid-state VAR compensators which can be
controlled using feedback signals. Theoretical studies show that
the lagging reactive power and the excitation voltage required can
be substantially reduced if the capacitive elements are connected
across the load, when the load supplied is inductive. The copper
losses due to the lagging current component which flows from the
excitation source to the load via the machine windings is then
reduced, enabling efficient conversion of mechanical energy to
electrical energy.
In practice, the generating scheme outlined in this chapter can
be implemented with more versatile features by incorporating
microcomputer control.
204
CHAPTER 7: SUMMARY AND CONCLUSIONS
7.1 Conclusions
The literature survey undertaken for this project indicates
that many different aspects of single-frame cascaded induction
machines have been investigated by previous researchers. The
contributions made in this thesis enhance the knowledge on the
analysis, design and operating characteristics of the SFCIM as a
special purpose machine.
The new steady-state circuit theory for those SFCIMs which have
a 'multicircuit single-layer bar rotor winding' uses the concept of
'cyclic' inductances, which had been used previously by
30 Poloujadoff . Assumptions made in the development of the new model 7 are similar to those made by Broadway and Burbridge and yet the new
model is relatively much simpler. Using the new model it has been
shown that the multicircuit rotor winding can be conveniently
modelled and the machine can be represented by a coupling impedance
matrix. Calculation of the elements of the coupling impedance
matrix is relatively straightforward. The presence of all unwanted
rotor space harmonics was represented in the coupling impedance
matrix by the use of 'cyclic' inductances and hence there was no
need to consider the leakage inductance components other than
overhang and slot-leakage inductances. All currents calculated
using the model are referred to their own windings and hence no
turns ratio calculations are required. The effects of saturation on
p- and q-pole pair airgap inductances and rotor cyclic inductances
205
were incorporated using saturation factors calculated independently
of each other. This is strictly not correct as the two fields are
15 simultaneously present. Broadway, Cook and Neal indicate that
work has been previously undertaken on the effects of saturation on
the operating characteristics of the SFCIM. However, it is felt
that a separate investigation is required on this aspect with
relevance to the new circuit theory model which has been developed.
Difficulties have been faced by previous researchers in the
experimental determination of steady-state equivalent circuit
parameters of the SFCIM. These are further aggravated when the
rotor winding is of the multicircuit type. When the rotor winding
is of the phase-wound type, the techniques described in Chapter 3
can be carefully applied to determine the parameters of the
equivalent circuit given. Despite the simplifying assumptions made
in the theory behind the experimental procedures for the estimation
of the parameters, good agreement was found between the predicted
results using the experimental equivalent circuit and the
experimentally observed characteristics. It is strongly recommended
that stabilised a.c. power supplies be used in future work with
regard to experimental determination of equivalent circuit
parameters. The magnitude of the effective turns ratio plays a
vital role in the calculation of the referred parameters of the
equivalent circuit. This ratio can be calculated in both tests
described in Chapter 3 and hence a check on its magnitude is
possible.
The investigation on iron losses undertaken in Chapter 4, while
enabling the calculation of core loss resistances in the equivalent
206
circuit, also describes the manner in which they are distributed in
the yoke and teeth of both stator and rotor, at standstill and
cascade synchronous speed of the SFCIM. The analysis is by no means
rigorous and not all components of iron losses have been considered.
The difficulties associated with modelling the hysteresis losses
over a range of speeds were stated. The theoretical and
experimental evidence shows that in asynchronous operation of the
SFCIM in the sub-synchronous range of speeds, the iron losses can be
maintained at a relatively low level when the large pole pair stator
side is supplied at constant frequency and the small pole pair
stator side is allowed to carry the slip-frequency currents. This
will also be true in doubly-fed operation where the SFCIM operates
at or below the cascade synchronous speed.
Some design aspects of phase-wound rotor windings were
considered in Chapter 5. For SFCIMs having wide ratio pole
combinations the phase-wound rotor windings are better suited than
the multicircuit single-layer bar rotor winding, as the coil pitch
and spread can be so selected that the performance can be optimised.
The investigations in Chapter 5 lead to the conclusion that the
primary aspect of importance is the minimisation of the harmonic
leakage inductance of the rotor winding. Although this cannot be
considered as a sufficient condition in the design of phase-wound
rotor windings, the other guiding principles considered in Chapter 5
will be of assistance in the design of SFCIMs. The overall
performance of the machine is governed finally by the properties of
the stator windings as well. The adoption of single stator
7 windings instead of separate 2p- and 2q-pole stator windings is
207
recommended wherever possible.
The steady-state operating characteristics of the SFCIM as a
variable-speed constant-frequency generator examined in Chapter 6
clearly illustrate that the resistance and leakage reactance
parameters play a vital role. In wind energy applications the speed
of the machine inevitably varies over a wide range and if the
generation has to continue providing a near stabilised output
voltage, the excitation voltage/current has to be controlled over a
wide range. Unless the total resistance of the machine is kept to a
minimum, the conversion efficiency can be quite low as in the
experimental machine. When the overall performance is considered,
it is felt that a modern permanent magnet excited a.c. generator
together with a solid-state rectifier/inverter scheme would perform
better in wind energy applications. For such a scheme the
power/weight ratio can be expected to be high compared to the scheme
considered in Chapter 6.
7.2 Suggestions for Future Research
The following suggestions are made for future research on
SFCIM.
(a) The coupling impedance matrix developed in Chapter 2 can be
further expanded to incorporate the coupling of rotor space
harmonics other than p- and q-pole pair fields onto the two
stator windings. This would involve inclusion of additional
stator to rotor mutual inductances which can be computed in a
manner similar to the inductances Mpr, Mqr, Mprm and Mqrm< The
respective slips [similar to s (= spsq)] can be worked out with
208
a knowledge of the direction of rotation of the space harmonics
of the rotor. As there are no applied voltages corresponding
to these harmonics the respective entries on the voltage vector
will be zeros.
Saturation and its effect on the performance of the SFCIM
have to be considered in relation to the new circuit theory
model and such a study would be a major research topic on its
own.
(b) Experimental procedures described for the determination of
equivalent circuit parameters can be applied to the axial flux
version of the SFCIM.
209
BIBLIOGRAPHY
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Wechselstrom-Treibmaschinen mit magnetischem Drehfelde zu
verringern', German Patent 73050, 1894.
3 HUNT, L.J. : 'A new type of induction motor', J. IEE,
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7. BROADWAY, A.R.W., and BURBRIDGE, L. : 'Self-cascaded machine: a
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Proc. IEE, Vol. 117, No. 7, July 1970, pp. 1277-1290.
8. BROADWAY, A.R.W., FONG, W., and RAWCLIFFE, G.H. : 'Improvements
in or relating to rotary electric machines', British Patent
Application 30814, 1966.
9. BROADWAY, A.R.W. : 'Cageless induction machine', Proc. IEE,
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10. GORGES, H. : 'Ueber Drehstrommotoren mit verminderter
Tourenzahl', Elektrotech. Z., Vol. 17, 1896, pp. 517-518.
210
11. SCHENFER, C. : 'Der Synchron-Induktionsmotor mit Erregung im
Stator', Electrotech. B Maschinenbau, Vol. 44, 1926,
pp. 345-348.
12. RUSSEL, R.L., and NORSWORTHY, K.H. : 'A stator-fed half-speed
synchronous motor', Proc. IEE, Vol. 104A, 1957, pp. 77-87.
13. BROADWAY, A.R.W., and TAN, S.C.F. : 'Brushless
stator-controlled synchronous-induction machine' , Proc. IEE,
Vol. 120, No. 8, August 1973, pp. 860-866.
14. COOK, B.J. : 'Self-cascaded alternator: a new form of high
frequency brushless generator', Ph.D. Thesis, University of
Bristol, 1972.
15. BROADWAY, A.R.W., COOK, B.J., and NEAL, P.W. : 'Brushless
cascade alternator', Proc. IEE, Vol. 121, No. 12,
December 1974, pp. 1529-1535.
16. SMITH, B.H. : 'Theory and Performance of a Twin Stator
Induction Machine', IEEE Trans., Vol. PAS-85, No. 2,
February 1966, pp. 123-131.
17. SMITH, B.H. : 'Synchronous Behaviour of Doubly Fed Twin Stator
Induction Machine', IEEE Trans., Vol. PAS-86, No. 10,
October 1967, pp. 1227-1236.
18. COOK, CD. : 'Dynamic characteristics of the doubly fed twin
stator induction machine', Ph.D. Thesis, University of New
South Wales, 1976.
19. COOK, CD., and SMITH, B.H. : 'Stability and stabilisation of
doubly-fed single-frame cascade induction machines',
Proc. IEE, Vol. 126, No. 11, November 1979, pp. 1168-1174.
211
SMITH, B.H., and COOK, CD. : 'An Incrementally Variable
Phase-Locked Control for a Polyphase Inverter', IEEE Trans.,
Vol. IM-27, No. 1, March 1978, pp. 74-76.
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Stabilisation of Single-Frame, Cascade Induction Machines as
Stepping Motors', CONUMEL 80, Int. Conf. on Numerical Control
of Electrical Machines, 1980.
NODA, J., HIRO, Y., and HORI, T. : 'Brushless Scherbius Control
of Induction Motors', IEEE Conf. Rec. of IAS, 9 th Annual
Meeting, Part I, 1974, pp. 111-118.
NONAKA, S., and OGUCHI, K. : 'Brushless Wound-Rotor Induction
Motor Driven by Self-Controlled Reversible Frequency
Converter', EEJ, Vol. 97, No. 2, 1977, pp. 40-48.
KUSKO, A., and SOMUAH, CB. : 'Speed Control of a Single-Frame
Cascade Induction Motor with Slip-Power Pump Back',
IEEE Trans., Vol. IA-14, No. 2, March/April 1978, pp. 97-105.
OGUCHI, K., and SUZUKI, H. : 'Speed Control of a Brushless
Static Kramer System', IEEE Trans., Vol. IA-17, No. 1,
January/February 1981, pp. 22-27.
ORTMEYER, T.H., and BORGER, W.U. : 'Brushless Generation with
Cascaded Doubly Fed Machines', Proc. IEEE National Aerospace
and Electronics Conference, 1983, pp.1420-1425.
ORTMEYER, T.H., and BORGER, W.U. : 'Control of Cascaded Doubly
Fed Machines for Generator Applications', IEEE Trans.,
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212
28. ORTMEYER, T.H. : 'Negative Frequency Aspects of Doubly Fed
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29. RIAZ, M. : 'Energy-Conversion Properties of Induction Machines
in Variable-Speed Constant-frequency Generating Systems',
Trans. AIEE, Vol. 78, Pt. II, March 1959, pp. 25-30.
30. POLOUJADOFF, M. : 'General Rotating mmf Theory of Squirrel Cage
Induction Machines with Non Uniform Airgap and Several Non
Sinusoidally Distributed Windings', IEEE Trans., Vol. PAS-101,
No. 3, March 1982, pp. 583-591.
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with rotor-bar and end-ring faults', Proc. IEE, Vol. 129,
Pt. B, No. 3, May 1982, pp. 93-100.
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M.E. Thesis, University of Melbourne, 1977.
33. ALGER, P.L. : 'Induction Machines', Gordon and Breach, 1970.
34. KESAVAMURTHY, N., and BEDFORD, R.E. : 'The Circuit Theory and
Calculations of Polyphase Induction Machines', IEE Monograph,
No. 304U, May 1958, pp. 499-508.
35. CHALMERS, B.J. : 'Electromagnetic Problems of A.C. Machines',
Chapman and Hall, 1965.
36. ALGER, P.L., and EKSERGIAN, R. : 'Induction Motor Core Losses',
J. AIEE, October 1920, pp. 906-920.
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Vol. 34, October 1915, pp. 2693-2720.
213
38. TRICKEY, P.H. : 'Iron-Loss Calculations on Fractional
Horsepower Induction Motors', Trans. AIEE, Vol. PAS-77,
Part III, February 1959, pp. 1663-1669.
39. Lysaght Electrical Steels Catalogue, John Lysaght (Australia)
Limited, 1973
214
APPENDIX 1: ROTOR CIRCUIT FLUX LINKAGES
1.1 Total Flux Linkage with a Cage Mesh due to Rotor Currents
ck From Equation (2.18) of Chapter 2, the total flux linkage, V ,,
with a cage mesh due to all rotor mesh currents is given by
P+q P+q z
^ck ~ LccXck + 2 cxcn + 2 Z ck,unm1unm n=l n=l m=l n*k
+ > M i , + 21 i , £ eum ukm ce ck m=l
+ V2ick-ic(k-l)-ic(k+l)) (A1>1)
where L = airgap inductance of a cage mesh cc
M = mutual airgap inductance between any two cage c
meshes
M = inductance of the end-ring section common to a cage eum th
mesh of a phase group and the m U-loop of the
same phase group
1 = end-ring inductaace of a cage mesh
= mutual airgap inductance between the cage mesh of
kth phase group and the mth U-loop of the nth phase
ce
M
group
1 = slot-leakage inductance of a bar b
215
and where the inductances L and M can be shown to be given by
^0 L = d - x ) a xo (A1.2) cc
g ^0
M = - x^ x^ a (A1.3) c c c
where g = airgap length
a = total airgap surface area of the rotor
airgap surface area of a cage mesh c ~ ~~~~ ~
According to the assumption made in Section 2.1 of Chapter 2 the
currents in all cage meshes form a balanced (p+q)-phase system,
p+q
i.e. ick+;> icn = 0 (A1.4)
n=l n*k
Using Equations (A1.2), (A1.3) and (A1.4), the first two terms of
Equation (Al.l) can be simplified as
P + Q Pr V . ^0 L i , + > M i = o x^ l , cc ck / c cn c ck n=l
n*k
= 1 i , (AL5) c ck
where 1 = -H- a xp (Al.6)
g
216
The inductance lc can be identified as the 'cyclic' inductance of a
cage mesh.
th The mutual inductance between a cage mesh and an m U-loop of
the k phase group is different from that between the same cage
mesh and an m U-loop of any other phase group. Thus, the 3 term
of Equation (Al.l) has to be split into two terms as
p+q z
1 1" ck,unm unm ~ / n=l m=l m=l
P+Q
M , , i . + X II, i ck,ukm ukm £ ck,unm unm
n=l n*k
where
M ck,ukm (1 - x ) x o
c um
M ck,unm x x o c um n*k
(A1.7)
(A1.8)
(A1.9)
and where airgap surface area of an m U-loop
um
The currents in all m U-loops in the rotor also form a balanced
(p+q)-phase system
i.e.
p+q
ukm Z un"!
n=l n*k
(ALIO)
217
Hence, Equation (A1.7) can be simplified as
P+q z z
L L ck,unm unm " 2 umXukm (Al.ll) n=l m=l m=i
^0 where 1 = o x U1 1JM
um um ^ Ai.i^ j g
The inductance l^ can be identified as the 'cyclic' inductance of
th II i
an m U-loop.
The cage mesh currents ig/^.^f ick and i . are displaced
by an electrical angle of 2i7p/(p+q) from each other with respect to
the 2p-pole field and hence
o • • .. . 2 lip ci . - l ,. ., - I ,, ,-. = 4i . sin — — -ck c(k-l) c(k+l) ck p+q
Thus the total flux linkage T , is given by CK
z z
lPr. :1 i , +\ li, +^> M i. ck c ck / um ukm /, eum ukm m=l m=l
(A1.13)
+ ^ b 'ck + "ae'ck ( A 1 - H )
where L, = 41, sin — B — bb b p+q
218
1.2 Total Flux Linkage with a U-loop due to Rotor Currents
From Equation (2.27) of Chapter 2, the flux linkages of the m
U-loop of the k phase group due to all rotor mesh currents is
given by
p+q
cn,ukm cn ck,ukm ck yr, = > M i + M , , i ukm £
n=l n*k
p+q
+ > M , i + L . , i i / ukm,unm unm ukm,ukm ukm n=l n*k
p+q z m
+2 2 Mukm,unh Xunh 2 eum lukh
n=l h=l h=l h*m
z
+ ^ Meuh ^kh + Meum ^k + 1oumiukm h=m+l
z
+ S M i ,_. + 21,i w_ (A1.15) 2 Moum,ouh Xukh + b1ukm
h=l h*m
219
In Equation (A1.15), the 1 and the 2 terms can be simplified
using Equations (A1.4), (A1.8) and (A1.9). Hence
p+q
M . i + M . , i . = 1 i . (A1.16) cn,ukm cn ck,ukm ck um ck
n=l n*k
In the 3 and the 4 terms of Equation (A1.15), the inductances
are given by
M . . = - — — x x a (A1.17) ukm,unm | um um
n*k S
L , = - i - (1 - x ) x o (A1.18) ukm,ukm um um
Using Equations (ALIO), (Al. 17) and (A1.18), the 3r< and the 4
terms of Equation (Al.15) can be simplified. Hence
p+q
1ukm,unm1unm T ^ukmtukm^ukm " xum xukm \ M i + I, = I i , (A1.19
n=l n*k
220
A. U
The 5 term of Equation (A1.15) can be simplified by splitting it
into different terms, i.e.
P+q z p+q m-l z
'ukmjunh^unh 2 2 Mukm,unh ^nh = } Ii ^km.unh^nh + 2 n=l h=l n=l h=l h=m+l h*m
(A1.20)
In Equation (A1.20) the inductance M', , represents the mutual ukm,unh *
inductance between the m U-loop and other U-loops which have a
smaller airgap surface area compared with the m U-loop. The f u
inductance M',' . represents the mutual inductance between the m ukm,unh
U-loop and other U-loops which have a larger airgap surface area
compared with the m U-loop. Each term on the right hand side of
Equation (A1.20) has to be further split into two terms, as the
mutual inductance between the m U-loop of the k phase group and
h U-loop of the same phase group is different from that between
the m U-loop of the k phase group and a h U-loop of a
different phase group. Thus, the right hand side of
Equation (A1.20) can be expanded as
p+q z m-l p+q
2 Z. Mukm,unh Xunh ~ 2 [ 1 Mukm,unhLunh + ukm,ukhXukh n=l h=l h=l n=l h*m n*k
z p+q
+ Z i Z Mukm,unh1unh + ukm,ukh\jkh h=m+l n=l n*k (A1.21)
221
The different inductances in Equation (A1.21) are given by
H0 Mukm,unh| =" xumxuh° <A1'22> n*k g
(A1.23) ukm,ukh
M' ' ukm,unh | n*k
:
~
^0 u (1 -
g
^0 - X
um g
Xuh) :
Xuh ° (A1.24)
^0 MV ,, = — — (1 - x ) x . a (A1.25) ukm,ukh um uh
, airgap surface area of an h U-loop where x , = - —
uh
Using these inductances Equation (A1.21) can be reduced as
p+q z m-l z
^ ^ M , uiu=^ 1 i,u+^> iv.1^ (A1.26)
Z Z ukm, unh unh Z um ukn Z uh ukh
n=l h=l h=l h=m+l h*m
^0 where 1 ^ = — a x^
222
By using the expressions given by Equations (A1.16), (A1.19), and
(A1.26) in Equation (A1.15) for ¥^km
m z m
^ukm = Uum + Meum) S k + 2 ^ ^kh + 2 Xuh Skh + Z Meum1ukh h=l h=m+l h=l
+ "\ M i + M ui ,u+ (21, +1 ) i u (AL27) 2 euh ukh Z oum.ouh ukh b oum ukm
h=m+l h=l h*m
223
APPENDIX 2: EXPRESSIONS FOR TORQUE
Expressions for torque of an SFCIM having a multicircuit
single-layer bar rotor winding with a (p+q)-bar cage and two U-loops
per cage mesh are derived in this appendix.
From Equation (2.53) of Chapter 2
1 (P+q)
r . - . S W . . . r S W
^ " "^ + Ck -^ L u p J L q
i i - I T * i _(rbb+2r )i ,
ck dt ck bb ce ck
-r ,i . ,i . - r „i , „i , (A2.1) eul ukl ck eu2 uk2 ck
st From Equation (2.54) of Chapter 2, for the 1 U-loop
1 Tp+qT
T p i f-ukl [
s w ., P P + T qi
ukl r s w
P P d J .2
1ukl~oTukl rulxukl
^eulWukl ' W u k f reu2iukliuk2 {klX
and for the 2 U-loop
_1 Tp+qT
rPi
uk2 P P | + T qi
uk2
rS W -,
P P «. n -> J
d r .2 Luk2~oT*uk2 ru2xuk2
"reu2ickiuk2 " reu2iuk21ukl ' reu2xuk2 .2 Lu)
(A2.3)
224
The expression for \? , given by Equation (2.19) of Chapter 2 when
expanded yields
i i-^n-^i = i ,(1 +lui+2l )-rrr-i i + i , (1 n+M .)—„-! ., ck dt ck ck c bb ce eft ck ck ul eul dt ukl
+ i . (1 0+M „)-4ri M (A2.4) ck u2 eu2 dt uk2
and the expression for ? . given by Equation (2.28) of Chapter 2
when expanded for m = 1 yields
^kinrjdfukl = Vl^ul^eu^'oT^k + V^ul-dT^kl
+ iukl1u2~dTiuk2 + iuklMeul"oT1ukl
d . M d . + xukl Meu2~oT1uk2 + 1ukloul,ou2~artuk2
+ i . ,(21,+1 J-i-i t1 (A2,5)
ukl b oul dt ukl
225
Carrying out a similar expansion for the second U-loop, i.e., m = 2
iuk2~St*uk2 = iuk2(1u2+Meu2)"aT1ck + 1uk21u2"aT1ukl
4. i 1 d i + i 1 d i Xuk2 u2~aT ukl uk2 u2~aT uk2
+ iuk2Meu2"aTiukl + iuk2Meu2~aT1uk2
+ i , „M uk2 ou2,oul~aT1ukl
+ iuk2 ( 2V 1ou2)^T iuk2 (A2.6
When the expressions given by Equations (A2.4), (A2.5) and
(A2.6) are averaged and added together, they sum to give a zero
average value. Therefore, by combining Equations (A2.1), (A2.2) and
(A2.3) and taking the average over a time period of the supply
frequency yields
* T - _P_±1 + T Tp+qJlPL p J <i
rs w p p
"• q J
- (r„ + 2r )-x bb ce
- r Tl
ul
22 22 {2 Xr2 p *rl . _ ^£2 " r " reul ~Z- eu2 2 u2
" r e u l V r l C ° S *rl " reu2IrIr2COS 4»r2
~ reu2IrlIr2COS(<l)rr<i)r2) (A2.7)
226
The right hand side of Equation (A2.7) can be shown to be equal to
the negative value of the total copper loss within a phase group of
the multicircuit rotor winding. When Equation (A2.7) is combined
with Equation (2.59) of Chapter 2 the component torques T and T
can be independently evaluated.
227
APPENDIX 3: THE WOUND ROTOR INDUCTION MACHINE AS A
CONSTANT-FREQUENCY VARIABLE-SPEED GENERATOR
Important operating characteristics of the SFCIM as a
constant-frequency variable-speed generator can be inferred from an
examination of the basic voltage equations of a conventional wound
rotor induction machine (WRIM). In relation to the WRIM, it is
assumed that the three-phase stator delivers constant frequency (w)
output voltage to the load and the three-phase rotor is supplied by
the variable-voltage variable-frequency (sw) source, where s is the
slip of the rotor with respect to the stator field.
Observing the form of Equations (2.36) and (2.61) of Chapter 2,
the steady-state coupled circuit equations of the WRIM can be stated
as
3 M Vj = [Rj +jw(L1- Ux)] Ix + jw I
(A3.1)
2 . 3 M T , = jw I. +
s 2 l
R, + jw (L2- M2) (A3.2)
where
R,
= per phase applied voltage on the stator
= per phase applied- voltage on the rotor
= per phase current of the stator
= per phase current of the rotor
= resistance of a phase winding of the stator
= self inductance of a phase winding of the stator
228
M = mutual inductance between any two phase windings of
the stator
M = maximum value of the mutual inductance between a
phase winding of the stator and a phase winding of
the rotor
R„ = resistance of a phase winding of the rotor
L„ = self inductance of a phase winding of the rotor
M„ = mutual inductance between any two phase windings of
the rotor.
Under open-circuit conditions on the stator, the current I. is
zero and hence the Thevenin's voltage source or the open-circuit
voltage (V ) is given by oc
oc JW
3 M
+ jw (L2 - M2)
(A3.3;
For the open-circuit voltage V to be constant in magnitude as oc
the slip changes, the condition to be satisfied is given by
V„.2
H ,R2,2
"• s J
w (L2 -M2) !A3.4)
where C is a constant
229
Equation (A3.4) suggests that as the magnitude of the slip s
increases, the voltage Vg has to increase in magnitude in order to
maintain a given open-circuit voltage constant.
The Thevenin's impedance or the short-circuit impedance Z of sc
the machine is given by
9 M2w2
Zsc = ' [ Rl + j w <L1 " M1>1 " R,
+ jw (L2 - M2)j
(A3.5)
The complexity of the short-circuit impedance given by
Equation (A3.5) does not permit any useful information to be
obtained without numerical work.
Multiplication of Equation (A3.1) by I. (complex conjugate of
L) and Equation (A3.2) by I„ (complex conjugate of I„), the
expressions for real and reactive power flows on a per phase basis
can be established. If the rotor consisted of a different number of
phases than the stator, these equations have to be appropriately
scaled using the number of phases on the stator and the rotor. Thus
Pl + l\ RX + l2 R2 (A3.6)
where P.. = per phase real power input to the stator
P„ = per phase real power input to the rotor
230
The reactive power flows are related by the expressi on
Q2 %l + = reactive power consumed by the magnetising and
s leakage reactances
(A3.7)
where ^ = per phase lagging reactive power input to the
stator
Q„ = Per phase lagging reactive power input to the rotor
Equation (A3.6) identifies the components of real power flow
which are well understood in the classical theory of induction
machines. Equation (A3.7) relating the components of reactive power
is comparatively less understood in doubly-fed operation. In the
operation of the WRIM as a constant-frequency variable-speed
generator, the phase-sequence of the variable-voltage
variable-frequency supply has to reverse at s = 0 and the sign
accompanied by Q„ also reverses at the same time if either Q. is
zero (which corresponds to a purely resistive load) or when a
lagging load is supplied, where Q. carries a negative sign.
However, the negative sign of Q„ in such a reversal should still be
interpreted to mean a lagging . reactive power required by the
machine. This anomaly with respect to doubly-fed operation of
induction machines is documented in Reference 28.
231
APPENDIX 4: HARMONIC CONTENT OF MMF WAVEFORMS PRODUCED BY
MULTICIRCUIT SINGLE-LAYER BAR ROTOR WINDING
This appendix presents the harmonic content of the mmf waveform
produced by the multicircuit single-layer bar rotor winding which
was used in the theoretical and experimental investigations in
Chapter 2. Two cases are presented where the SFCIM is supplied on
the 2- and 6-pole stator sides respectively. The input operating
conditions are as per Figures 2.7 and 2.9 of Chapter 2. In
Tables A4.1 and A4.2, the 2-pole mmf wave has been considered as the
reference wave.
232
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