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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>

o Q. I

cx cu

CL 3

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

... i<? N

C |

£ J -w+

1

/«-

3 u

1

O 2 l-H > < EC O ta

w H

ta O CO

H I — I

a

ta

o H O >-<C J

Csl

w

M ta

a z i — i

o

o O « OS

pa

«

w < j i

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

LO

PH hH J cn 5C H hH SK

H 2 PH PH PH ^ a PH

o 2 O hH H <; hH PH < >

CM l-H

CM

PJ

g C5 h-1 Ch

N PC o m

P cd

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• ^ ^ « -

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P cd

VS CD •H

rH ft ft 3 cn CH O P cd p cn cu rH

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O

P u o si cn (H O P cd P cn

0) r-H O ft 1 CM

s •

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ts • cn

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ctn iN3yyn3

67

si

C\J

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(J DC

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ts •

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in si m si

a o 2 PJ P3 P5 53 U P3 O H < H Kl PJ J

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PJ

B CJ l-l Ph

((U) iN3yyn3 yoiuis 310CJ-9}1"!

ts

68

Q_

CO

PH

hH •J C/J

SC E-i

PJ 53 <3 PJ O H PH

o 2 O hH

H < hH PH

< > CM

PJ

o PH

(WN) anoyoi

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

oov

ro in

cn « — i

in s +

oo cn

CD oo

N 0 ^

co OO CO cn cn oo

OO CD

CO •

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m in

"3-

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tn ca ou

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ca ca (VJ

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f— 1—1

CO CD n DC CT _J ZD LD 2 cr cc o i— n DC

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l-H CO

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w O O

« H r-i

Sc c

OH O -2H

< r

PH

UH

o O i — i

<

>

•

M

w !=3

(D - OJ a co io OU OU OJ OJ r-i r-H

(M) e d QNbJ Td

^ co oo ca ou ^

I I I

85

oo onooro • • a i a

^r r-casn -^

oo .

r-l

LO

OO CD • a

CD CO •**• r

CO a

r-t-

oo •

t-t

"3-ca ou

ro ia ou

ou ca ou

ca ou

ca ca ou

CD CD

1 ta

i

ca ca

1 ca

•

ca CD

1 ca

•

ca oo

1 ca a

ca r-

1 ca • ca to

1 ca a

ca LO

1 ca • ca •t

1 ta a

ca CO

1 ca

•

ca ou

1 ca e

a

ca T—«

00 CD

3r-4

a

(UA) eS QNU TS

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

r~ oo ro co C D CO OO

to ro --. CD oo r- •<* w m m ~<t •*? *<t **

ca ou

0<l +O 4-K«

+

ro ca ou

ou ca ou

ca ou

ca ca ou

CD CD

oo (D

CSUi I

CD

ta i

TI­

CS •

ro ca a

CM

ca ca •

ca

ca

CD

ca •

oo

ca

(M) loi

88

r- oo co oo CD co oo

to co *-i CD oo r- "«t LO LO LO Tf NJ- -vf Tf

o o +o<f-txx

r-LO

CD LO

LO LO

Tt LO

CO LO

OU LO

LU LD CE

CD

w o < H o > o H < H CO

W -J

o O H

I to 33 E-i

LO

ca LO

CD Tt

CO Tf

r-

az o \— CE f— CO

LU _J O ID-

CO

+->

OH

C»H

O 2 O i — i

< h-I

< >

a

w

CD Tt

C5 M lo­

in Tt

Tf Tt

ca •

Tf

ca •

CO

ta •

ou

ta ca

ta

ta •

CD

ta •

oo

ta a

r-

(M) }°1 UJ

ta a

CD

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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G *^*> X

«" - -«

cT _*. CM

X

'a . -CM

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X! ^.^ a r-H

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«* r—i

CM

4->

cn <u H

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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cc o Q CC

o

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cr

2 LU DC DC

cu •

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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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1

L

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LJ

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is O Cb Q < O J

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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)

Q < O J

K E-< hH * CO LU CO CO

o -J ca w PH

0-o u Cb

o

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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5873401897.

2. Siemens and Halske : 'Verfahren, die Umdrehungszahl von

Wechselstrom-Treibmaschinen mit magnetischem Drehfelde zu

verringern', German Patent 73050, 1894.

3 HUNT, L.J. : 'A new type of induction motor', J. IEE,

Vol. 39, 1907, pp. 648-667.

4. HUNT, L.J. : 'The "cascade" induction motor', J. IEE,

Vol. 52, 1914, pp. 406-426.

5. CREEDY, F. : 'Some developments in multi-speed cascade

induction motors', J. IEE, Vol. 59, 1921, pp. 511-532.

6. BROADWAY, A.R.W., and THOMAS, G. : 'Single-unit p.a.m.

induction frequency converters', Proc. IEE, Vol. 114, No. 7,

July 1967, pp. 958-964.

7. BROADWAY, A.R.W., and BURBRIDGE, L. : 'Self-cascaded machine: a

low-speed motor or high-frequency brushless alternator',

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,

Vol. 118, No. 11, November 1971, pp. 1593-1600.

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.

SMITH, B.H., COOK, CD., and PERERA, B.S.P. : 'Operation and

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.,

Vol. PAS-103, No. 9, September 1984, pp. 2564-2571.

212

28. ORTMEYER, T.H. : 'Negative Frequency Aspects of Doubly Fed

Machine Analysis', Proc. IEEE, Vol. 71, No. 8, August 1983,

p. 1017.

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.

31. WILLIAMSON, S. : 'Steady-state analysis of 3-phase cage motors

with rotor-bar and end-ring faults', Proc. IEE, Vol. 129,

Pt. B, No. 3, May 1982, pp. 93-100.

32. GOL, 0. : 'Induction Machine Models for Design and Analysis',

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.

37. BALL, J.D. : 'The Unsymmetrical Hysteresis Loop', Trans. AIEE,

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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