7373941 010 High Speed Solid Rotor Induction Motor (1)

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Lappeenrannan teknillinen yliopistoLappeenranta University of Technology

Jussi Huppunen

HIGH-SPEED SOLID-ROTOR INDUCTION MACHINE ELECTROMAGNETIC CALCULATION AND DESIGN

Thesis for the degree of Doctor of Science

(Technology) to be presented with due

permission for public examination and

criticism in the Auditorium 1382 at

Lappeenranta University of Technology,

Lappeenranta, Finland on the 3 rdofDecember, 2004, at noon.

Acta UniversitatisLappeenrantaensis197


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ISBN 951-764-981-9ISBN 951-764-944-4 (PDF)

ISSN 1456-4491

Lappeenrannan teknillinen yliopisto

Digipaino 2004


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ABSTRACT

Jussi Huppunen

High-Speed Solid-Rotor Induction Machine Electromagnetic Calculation and Design

Lappeenranta 2004

168 p.Acta Universitatis Lappeenrantaensis 197Diss. Lappeenranta University of TechnologyISBN 951-764-981-9, ISBN 951-764-944-4 (PDF), ISSN 1456-4491.

Within the latest decade high-speed motor technology has been increasingly commonly applied

within the range of medium and large power. More particularly, applications like such involved

with gas movement and compression seem to be the most important area in which high-speed

machines are used.

In manufacturing the induction motor rotor core of one single piece of steel it is possible to

achieve an extremely rigid rotor construction for the high-speed motor. In a mechanical sense,

the solid rotor may be the best possible rotor construction. Unfortunately, the electromagnetic

properties of a solid rotor are poorer than the properties of the traditional laminated rotor of an

induction motor.

This thesis analyses methods for improving the electromagnetic properties of a solid-rotor

induction machine. The slip of the solid rotor is reduced notably if the solid rotor is axiallyslitted. The slitting patterns of the solid rotor are examined. It is shown how the slitting

parameters affect the produced torque. Methods for decreasing the harmonic eddy currents on

the surface of the rotor are also examined. The motivation for this is to improve the efficiency

of the motor to reach the efficiency standard of a laminated rotor induction motor. To carry out

these research tasks the finite element analysis is used.

An analytical calculation of solid rotors based on the multi-layer transfer-matrix method is

developed especially for the calculation of axially slitted solid rotors equipped with well-

conducting end rings. The calculation results are verified by using the finite element analysis

and laboratory measurements. The prototype motors of 250 300 kW and 140 Hz were tested


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to verify the results. Utilization factor data are given for several other prototypes the largest of

which delivers 1000 kW at 12000 min-1.

Keywords: high-speed induction machine, solid rotor, multi-layer transfer-matrix, harmoniclosses.

UDC 621.313.333 : 621.3.043.3


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Acknowledgements

In 1996, at the Laboratory of Electrical Engineering, Lappeenranta University of Technology,

the research activities related to this thesis got started, being part of the project Development

of High-Speed Motors and Drives. The project was financed by the Laboratory of Electrical

Engineering, TEKES and Rotatek Finland Oy.

I wish to thank all the people involved in the process of this thesis. Especially, I wish to express

my gratitude to Professor Juha Pyrhnen, the supervisor of the thesis for his valuable comments

and corrections to the work. His inspiring guidance and encouragement have been of enormous

significance to me.

I wish to thank Dr. Markku Niemel for his valuable comments. I also thank the laboratory

personnel Jouni Ryhnen, Martti Lindh and Harri Loisa for their laboratory arrangements. I am

deeply indebted to all the colleagues at the Department of Electrical Engineering of

Lappeenranta University of Technology and at Rotatek Finland Oy for the fine and challengingworking atmosphere I had the pleasure to be surrounded with.

I am deeply grateful to FM Julia Vauterin for revising my English manuscript.

I also thank the pre-examiners Professor Antero Arkkio, Helsinki University of Technology,

and Dr. Jouni Ikheimo, ABB Motors.

Financial support by the Imatran Voima Foundation, Finnish Cultural Foundation, South

Carelia regional Fund, Association of Electrical Engineers in Finland, Walter Ahlstrm

Foundation, Jenni and Antti Wihuri Foundation, Teknologiasta Tuotteiksi Foundation and TheGraduate School of Electrical Engineering is greatly acknowledged.

Most of all, to Maiju, Samuli and Julius: Your simple childs enthusiasm and your laugh gave

me strength and kept me smiling. I am indebted to Saila for her love and patience during the

years. Finally, my dear friends, without your warm support, endless patience and belief I would

never have roamed this far.

Lappeenranta, November 2004. Jussi Huppunen


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Contents

ABBREVIATIONS AND SYMBOLS .........................................................................................9

1. INTRODUCTION .......................................................................................... .....................15

1.1 APPLICATIONS OF HIGH-SPEED MACHINES .....................................................................18

1.2 HIGH-SPEED MACHINES ..................................................................................................20

1.3 SOLID-ROTOR CONSTRUCTIONS IN HIGH-SPEED INDUCTION MACHINES ........................22

1.4 OBJECTIVES OF THE WORK.............................................................................................27

1.5 SCIENTIFIC CONTRIBUTION OF THE WORK......................................................................28

1.6 OUTLINE OF THE WORK..................................................................................................30

2. SOLUTION OF THE ELECTROMAGNETIC FIELDS IN A SOLID ROTOR .......31

2.1 SOLUTION OF THE ELECTROMAGNETIC ROTOR FIELDS UNDER CONSTANT PERMEABILITY34

2.2 CALCULATION OF A SATURATED SOLID-ROTOR.............................................................41

2.2.1 Definition of the fundamental permeability in a non-linear material ..................452.2.2 Rotor impedance....................................................................................................46

2.3 EFFECTS OF AXIAL SLITS IN A SOLID ROTOR...................................................................47

2.4 END EFFECTS OF THE FINITE LENGTH SOLID ROTOR.......................................................49

2.4.1 Solid rotor equipped with high-conductivity end rings........................................49

2.4.2 Solid rotor without end rings.................................................................................52

2.5 EFFECT OF THE ROTOR CURVATURE...............................................................................57

2.6 COMPUTATION PROCEDURE DEVELOPED DURING THE WORK........................................59

3. ON THE LOSSES IN SOLID-ROTOR MACHINES.....................................................62

3.1 HARMONIC LOSSES ON THE ROTOR SURFACE .................................................................63

3.1.1 Winding harmonics ......................................................................................... ......63

3.1.2 Permeance harmonics............................................................................................69

3.1.3 Decreasing the effect of the air-gap harmonics....................................................76

3.1.4 Frequency converter induced rotor surface losses................................................86

3.2 FRICTION LOSSES............................................................................................................87

3.3 STATOR CORE LOSSES ....................................................................................................90

3.3.1 Stator lamination in high-speed machines............................................................94

3.4 RESISTIVE LOSSES OF THE STATOR WINDING .................................................................94

3.5 LOSS DISTRIBUTION AND OPTIMAL FLUX DENSITY IN A SOLID-ROTOR HIGH-SPEEDMACHINE ........................................................................................................................96

3.6 RECAPITULATION OF THIS CHAPTER..............................................................................97


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4. ELECTROMAGNETIC DESIGN OF A SOLID-ROTOR INDUCTION MOTOR ..99

4.1 MAIN DIMENSIONS OF A SOLID-ROTOR INDUCTION MOTOR...........................................99

4.1.1 Utilization factor....................................................................................................99

4.1.2 Selection of theL/D-ratio....................................................................................103

4.1.3 Slitted rotor with copper end rings......................................................................104

4.1.4 Effects of the end-ring dimensions .....................................................................108

4.2 DESIGN OF SLIT DIMENSIONS OF A SOLID ROTOR.........................................................109

4.2.1 Solving the magnetic fields of a solid-rotor induction motor by means of theFEM-analysis.......................................................................................................110

4.2.2 FEM calculation results.......................................................................................115

4.2.3 Study of the rotor slitting ....................................................................................119

4.2.4 Comparison of the FEM with the MLTM method .............................................127

4.3 MEASURED RESULTS ....................................................................................................135

4.4 DISCUSSION OF THE RESULTS .......................................................................................136

5. CONCLUSION ......................................................................................... .........................138

5.1 DISCUSSION..................................................................................................................138

5.2 FUTURE WORK..............................................................................................................139

5.3 CONCLUSIONS ..............................................................................................................140

REFERENCES: ................................................................................................. ........................143

APPENDIX A.............................................................................................................................153

APPENDIX B ...................................................................................................... .......................155

APPENDIX C.............................................................................................................................162

APPENDIX D.............................................................................................................................164

APPENDIX E ...................................................................................................... .......................166


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9

Abbreviations and symbols

Roman letters

a abbreviation, function, number of parallel conductors, constant

a1k factor for calculating the slot harmonic amplitudes

A area, linear current density, vector potential

Aj cross-section area of one conductor

A magnetic vector potential (vector)

b flux density, function, distance

B magnetic flux density

Bn magnitude of magnetic flux density drop

c function, constant

C constant, utilization factor

CT torque coefficient

d function

dk thickness of layer

dp penetration depth

dc diameter of conductor

D diameter, electric flux density

E electric field strength, electromotive force (emf)

Eew distance of the coil turn-end

f frequency

F function

g boundary of regionG complex constant

H magnetic field strength

I current, modified Bessel function

J current sheet

J current density

k number of layer, factor, function, coefficient

k1 roughness coefficient

k2 velocity factor

kC Carter factor

K number of layers, function, modified Bessel function

K0 constant


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KC curvature factor

Ker end-effect factor

l length

lm length of one turn of the winding

L length

L electrical length

m number of phases

n constant, number of coil turns in one slot

N number of turns in series per stator phase

o width of slot opening

n unit normal vector

p pole pairs, power

P active power

q number of slots per phase and pole

qm mass flow rate

Q function

QR number of rotor slitsQS number of stator slots

r rotor radius

r rotor radius vector

R resistance

Rea Reynolds number of axial flow

Rer tip Reynolds number

Re Couette Reynolds number

S apparent power, surface

S Poynting vector, Surface vector

S complex Poynting vector

s slip

t time, thickness, width

T torque

Tk transfer matrix of layerk

u function, peripheral speed of the rotor

U voltage

v number of harmonic order, volume

V volume

vm mean axial flow velocity

w width

W energy


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x function

x, y, z coordinates

X reactance

Yk complex function of layerk

Z impedance

Greek letters

factor, end-effect factor, angle

complex function

flux distortion factor

factor

complex function, a measure of field variation in the axial direction

air-gap length

temperature coefficient of resistivity, permittivity

function

angle

magnetomotive force (mmf)

magnetic conductance

complex function of slip associated with penetration depth

permeability, dynamic viscosity of the fluid

0 permeability of vacuum

r relative permeability

efficiency, packing factor

winding factor

resistivity, charge density, mass density of the fluid, material density

conductivity, material loss per weight

Maxwell's stress tensor

leakage factor

lamination thickness

p pole pitch

u slot pitch

magnetic flux

chord factor

s stator angular frequency

mechanical rotating angular speed


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Subscripts

ave average

c cylindrical shell region, conductor

C Carter

Cu copper

class classicaldyn dynamic

e electric

ec eddy current

em electromagnetic

er end region

exc excess

Fe iron

fr friction

i index

in input

harm harmonic

hys hysteresis

k layer

lin linear

m magnetic

max maximum value

mech mechanical

min minimum value

R rotor

s supply, synchronous

S statorsl slip

sw switching

t tooth

tot total

u slot, slit

v harmonic of orderv

x,y,z coordinates

air-gap

0 basic value, initial value

1 fundamental, bottom layer


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Superscripts

R rotor

S stator

Other notations

a magnitude ofa

a complex form ofa

a vectora (inx,y,zcoordinates)

a complex form of vectora (time-harmonic presentation)

a peak value ofa

Acronyms

AC alternating current

emf electromotive force

DC direct current

FEM finite element method

IGBT insulated gate bipolar transistor

IM induction machine

MLTM multi-layer transfer-matrix

mmf magnetomotive force

PMSM permanent magnet synchronous machine

PWM pulse width modulation

SM synchronous machine


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1. Introduction

It is due to the remarkable development in the field of frequency converter technology that it

has become feasible to apply the variable speed technology of different AC motors to a wide

range of applications. There exists a growing need for direct drive variable speed systems.

Direct drives do not require reducing or multiplier gears, which are indispensable in

conventional electric motor drive systems. The use of direct drives is economical in both energy

and space consumption, and direct drives are easy to install and maintain. Traditionally, if the

motor drive should produce high speeds, multiplier gears are used.

There are several definitions for the term high-speed. In some occasions, the high speed is

determined by the machine peripheral speed. This can be justified from the mechanical

engineering point of view. Speeds over 150 m/s are considered to be high speeds (Jokinen

1988). This kind of a peripheral speed may, however, be reached with a two-pole, 50 Hz

machine which has a rotor diameter of 0.96 m. An electrical engineer may not regard a 50 Hz

machine as a high-speed machine. From the motor manufacturers point of view a two-pole

machine the supply frequency of which is considerably higher than the usual 50 Hz or 60 Hz is

normally considered to be a high-speed machine. However, some motor manufacturers have

called large 3600 min-1 machines high-speed machines. The difference of terms used in the

subject can be explained from the other viewpoint, which is that of the power electronics.

Present-day frequency converters are well able to produce frequencies up to a few hundreds of

hertz. However, the voltage quality of many converters is no more satisfactory if a purely

sinusoidal motor current is required. With respect to the present-day high-power IGBT-

technology the switching frequency is limited typically to 1.5 6 kHz. Lhteenmki (2002)

shows that the frequency modulation ratio (fsw/fs) should be at least 21 in order to succeed in

producing good quality current for the motor. It might thus be calculated that, as present-day

industrial frequency converters are considered, frequencies in the range of 100 400 Hz

appear to be high frequencies. There are several research projects aiming at the design of ultra

high-speed machinery. For example, Aglen (2003) reported the application of an 80000 min-1

rotating permanent magnet generator to a micro-turbine and Spooner (2004) described the

project the objective of which was the design of a 6 kW, 120000 min-1 axial flux induction

machine to be applied to a turbo charger. This thesis, however, focuses on electric machines


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that run at moderate speeds and with moderate power. The motor supply frequencies vary

between 100 Hz and 300 Hz and the motor powers between 100 ... 1000 kW.

The idea of using high-speed machines, which are rotating at higher speeds than it would be

possible to directly reach by means of the network frequency, is to replace a mechanical

gearbox by an electrical one and attach a load-machinery directly on the motor shaft. This gives

also full speed control for the drive. The use of converters has become possible in the latest

decades as high switching frequency voltage source converters often known as inverters

have came into the market. Converters, however, cause extra heating problems even in normal

speed machines and thus a careful design combining the inverter with a solid-rotor machine is

needed.

The technology research in the field of high-speed machines has been particularly active in

Finland. Pyrhnen (1991a) studied ferromagnetic core materials in smooth solid rotors.

Lhteenmki (2002) researched rotor designs and voltage sources suitable for high-speed

machines. His study focused on the design of squirrel cage and coated solid rotors. Saari (1998)

studied thermal analysis of high-speed induction machines and Kuosa (2003) analysed the air-

gap friction in high-speed machines. Antila (1998) and Lantto (1999) studied active magnetic

bearings used in high-speed induction machines. However, all of the above-mentioned studies

concentrated on machines running faster than 400 Hz. This thesis focuses on machines that run

at supply frequencies from 100 Hz to 300 Hz.

Also some other dissertations treating the solid rotor have been done. Peesel (1958) studied

experimentally slitted solid rotors in a 19 kW, 50 Hz, 4-pole induction motor. He manufactured

and tested 25 different rotors. Dorairaj (1967a; b; c) made experimental investigations on the

effects of axial slits, end rings and cage winding in a solid ferromagnetic rotor of a 3 hp, 50 Hz,

6-pole induction motor. Balarama Murty (Rajagopalan 1969) also studied the effects of axial

slits on the performance of induction machines with solid steel rotors. Wilson (1969) introduced

a theoretical approach to find out which is the impact of the permeability of the rotor material

on a 5 hp, 3200 Hz solid-rotor induction motor. Shalaby (1971) compared harmonic torques

produced by a 3.6 kW, 50 Hz, 4-pole induction machine with a laminated squirrel-cage rotor

and by the same machine with a solid rotor. Woolley (Woolley 1973) examined some new

designs of unlaminated rotors for induction machines. Zaim (Zaim 1999) studied also solid-

rotor concepts for induction machines.


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The laboratory of electrical engineering at Lappeenranta University of Technology (LUT) has

an over two decades long experience in and knowledge about the design and manufacturing of

high-speed solid-rotor induction motors. During the latest years research has been focused on

the improving of the efficiency of the high-speed solid-rotor motor construction. It has turned

out that, when a solid rotor is used, it is extremely important to take care of the flux density

distribution on the rotor surface. A perfectly sinusoidal rotor surface flux density distribution

produces the lowest possible losses. This is valid for both time dependent and spatial

harmonics. Because even a smooth solid construction high-speed steel rotor runs at quite a low

per-unit slip, this indicates that it is possible to reach a good efficiency if the stator losses and

the harmonic content on the air-gap flux and the rotor losses are kept low. Research has given

good results and the efficiencies of the high-speed motors have increased up to the level of the

efficiencies of typical 3000 min-1 commercial induction motors of the same output power.

At LUT, research in the field got started with the study on a 12 kW, 400 Hz induction machine

(Pyrhnen 1991a). Later, the properties of the machine were improved by means of a new stator

design and by using different rotor coatings and end rings (Pyrhnen 1993). After the promising

research results, 16 kW, 225 Hz induction motor structures with a smooth, a slitted and a

squirrel-cage solid rotor were tested for milling machine applications (Pyrhnen 1996). Later, 8

kW, 300 Hz and 12 kW, 225 Hz copper squirrel-cage solid-rotor induction motors were

manufactured to be used in milling spindle machines.

The next stage brought the investigation of bigger machines. A 200 kW, 140 Hz slitted solid-

rotor induction machine and a 250 kW, 140 Hz slitted solid-rotor induction machine with

copper end rings were analyzed (Huppunen 1998a). Afterwards, several induction machines

with both rotor types in the power range of 150 kW 1000 kW and in the supply frequency

range of 100 200 Hz were designed, manufactured and tested in co-operation with Rotatek

Finland Oy and LUT.

LUT has also cooperated in the developing of some permanent magnet high-speed machines.

Permanent magnet machines with output powers and rotational speeds of 20 kW, 24000 min-1

and 400 kW, 12000 min-1 (Pyrhnen 2002) were designed at LUT. Permanent magnet high-

speed machines have, however, several manufacturing related disadvantages and, therefore, this

machine type has not yet become popular for production in medium and large power range.


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Contrarily to this, the simple, rugged solid-rotor high-speed induction machine seems to be an

attractive solution for several industrial applications even though its efficiency is somewhat

lower and the size somewhat larger than the corresponding values of a PMSM at the same

performance.

Generally, the output torque of an electric machine is proportional to the product of the ampere-

turns and the magnetic flux per pole. Since the ampere-turns and the magnetic flux per pole

have limited values for a given motor size, the most effective way to increase the output power

is to drive the machine at a higher speed than normally.

The main advantages of using the motor in a high-speed range are the reduction of the motor

size and the absence of a mechanical gearbox and mechanical couplers. When using appropriate

materials the volume per power ratio and the weight per power ratio are nearly inversely

proportional to the rotating speed in the high-speed range. Thus, when the motor speed is near

10000 min-1, the motor size and the weight will decrease depending on the cooling

arrangements to about one third of the size of a conventional network frequency motor for

3000 min-1. This is valid for open motor constructions. If a totally closed construction is used

the benefit of the reduced motor size is lost.

Solid-rotor constructions are used because of mechanical reasons. This rotor type is the

strongest possible one and may be used in conjunction even with mechanical bearings at

elevated speeds since the rotor maintains its balance extremely well. When the load is directly

attached onto the solid-rotor shaft and elevated speed is used, the solid-rotor construction is still

able to achieve a sufficient mechanical strength and avoid balance fluctuations and vibrations,

which might damage the bearing system.

1.1 Applications of high-speed machines

High-speed solid-rotor induction motors may be used in power applications ranging from a few

kilowatts up to tens of megawatts. The main application area lies in the speed range where

laminated rotor constructions are not rigid enough as the mechanical viewpoint is considered.

Jokinen (1988) defined the speed limits for certain rotor types. The curves in Fig. 1.1 are

obtained, when conventional electric and magnetic loadings are used, the rotors are

manufactured of steel with a 700 MPa yield stress and the maximum operating speed is set 20

percent below the first critical speed. The rotational speed limit for the laminated rotors varies


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from ca. 50 000 min-1 to 10 000 min-1 while the power increases from a few kilowatts to the

megawatt range. However, this speed level may demand several special constructions e.g. rotors

with no shaft and with FeCo-lamination as well as with CuCrZr-alloy bars. Also the upper

speed limit for the solid-rotor technology is set by the mechanical restrictions and is 100 000

min-1 to 20 000 min-1, respectively. But, these mechanical restrictions define the maximal speed

for a certain rotor volume. The limiting power, however, is always defined by the thermaldesign of the machine.

10

100

1000

10000

1000 10000 100000

Rotational speed [rpm]

Maximumpower[kW]

Laminated

rotor

Solid rotor

Fig. 1.1. Powers limited by the rotor material yield stress (700 MPa) versus rotational speed (Jokinen

1988).

High-speed machines are mainly applied to blowers, fans, compressors, pumps, turbines and

spindle machines. The best efficiencies for these devices are achieved at elevated speeds, and

by using high-speed machines gearboxes and couplings can be avoided. The biggest potential

for high-speed machines lies on the field of turbo-machinery. Potential applications are blowers,

fans, gas compressors and gas turbines, because the rotational speeds of the gas compression

units are typically high. A common way to manufacture a gas compression unit is to use a

standard electric motor and a speed-increasing gearbox. Such machinery is manufactured by

Atlas Copco, Dresser-Rand, Solar Turbines, MAN Turbo, etc. During the latest decades high-

speed machines have been pushed on the market as an interesting solution to increase the total

system efficiency and to minimise total costs.


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Until the mid-1980s, the load commutated thyristor inverter for synchronous machines was the

only viable option for medium voltage, megawatt power range electric adjustable speed control.

Thus, synchronous motors made up the vast majority of all large high-speed installations before

1990. Since the mid-1980s, reliable electric adjustable speed control has been available for

medium voltage, megawatt-range, induction motors. As the acceptance of the induction motor

control technology in industry increased, it was only consequent that this technology wasconsidered to be applied also to high-speed use (Rama 1997).

1.2 High-speed machines

There are mainly two types of high-speed machines on the present-day market: High-speed

induction machines and high-speed synchronous machines with permanent magnet excitation.

However, minor research of claw-pole synchronous, synchronous reluctance and switched

reluctance high-speed machines is done as well. When the speed is high, centrifugal forces and

vibrations play an important role. Firstly, the rotor must have sufficient mechanical strength to

withstand centrifugal forces. Secondly, the designer must take the natural frequencies of the

construction into account. The critical frequencies may be handled in two ways; either the rotor

is driven under the first critical speed, which needs a strong construction and thick shafts, or the

rotor is driven between critical speeds. The latter obviously reduces the operating speed range

into a narrow speed area.

In induction machine applications - as far as the peripheral speed of the rotor is low enough, and

thus the mechanical loading is not a limiting factor - the laminated rotor with a squirrel-cage is

widely used. The first critical speed of this rotor type tends to be much lower than that of a solid

rotor. When the mechanical loading is heavy, solid-rotor constructions are used. Also in

permanent magnet rotors the laminated constructions with buried magnets can be used if the

mechanical stiffness of the shaft permits it. When the peripheral speed of a PMSM is high, a

solid steel rotor body is used and a magnet retaining ring or sleeve is needed. The retaining ring

is usually made of glass or carbon fibres, or of some non-ferromagnetic steel alloy material.

The issue of the state-of-the-art high-speed technology may be covered by making an analysis

of the articles dealing with the subject and an examination of the data sheets of the motor

manufacturers. Table 1.1 lists some high-speed electric machines that were selected from the

result of a literature search and table 1.2 gives some high-speed electric machine manufacturers.

The trend seems to be that for high-speed motors with power larger than 100 kW the induction


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motor type is commonly used and in smaller power ranges also the permanent magnet machine

type is used. Another conclusion might be that large natural gas pumping high-speed

applications in the megawatt range (Rama 1997) do exist and also small power applications

seem to be surprisingly general. Applications in the low voltage middle power range between

100 kW and 1000 kW and above 10000 min-1 are rarely used.

Table 1.1. Some high-speed electric machines selected from literature.

Power/kW Speed/min-1

Motor type Reference:

41000 3750 Synchronous motor Rama (1997), gas compressor

38000 4200 Synchronous motor Kleiner (2001), gas compressor

13000 6400 Synchronous motor Steimer (1988), petrochem. application

11400 6500 Synchronous motor Lawrence (1988), gas compressor

10000 12000 Solid-rotor IM, caged Ahrens (2002), prototype

9660 8000 Induction motor Rama (1997), gas compressor

9000 5600 Synchronous motor Khan (1989), feed pump

6900 14700 Laminated-rotor IM McBride (2000), gas compressor

6000 10000 Laminated-rotor IM Gilon (1991), gas compressor

5220 5500 Solid-rotor IM, caged LaGrone (1992), gas compressor

2610 11000 Solid-rotor IM, caged Wood (1997), compressor

2300 15600 Solid-rotor IM, caged Odegard (1996), gas compressor

2265 12000 Induction motor Rama (1997), pump

2000 20000 Induction motor Graham (1993), gas compressor

1700 6400 Induction motor Mertens (2000), chemical compressor

270 16200 Laminated-rotor IM Joksimovic (2004), compressor

250 8400 Solid-rotor IM, endrings

Huppunen (1998a), blower

200 12000 Solid-rotor IM, caged Ikeda (1990), prototype

131 70000 Permanent magnet SM Bae (2003), micro-turbine

110 70000 Permanent magnet SM Aglen (2003), micro-turbine

65 30500 Coated

Solid rotor IM, caged

Laminated-rotor IM

Lhteenmki (2002), prototypes

62 100000 Coated solid-rotor IM Jokinen (1997), prototype

60 60000 Coated solid-rotor IM Lhteenmki (2002), prototype

45 92500 Induction Motor Mekhiche (1999), turbo-charger

40 40000 Permanent magnet SM Binder (2004), prototype

30 24000 Permanent magnet SM Lu (2000), prototype


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22 47000 Permanent magnet SM Mekhiche (1999), air condition

21 47000 Laminated rotor IM Soong (2000), cooling compressor18

12

13500

13500

Solid-rotor IM, caged

Solid-rotor IM

Solid, slitted-rotor IM

Pyrhnen (1996), milling machine

11 56500 Laminated Kim (2001), compressor

Table 1.2. High-speed stand-alone electric motor manufacturers in the power range over 100 kW.

Power range/kW Speed range/ min-1 Rotor type Manufacturer

1000 25000 6000 18800 Induction Alstom

30 1500 20000 90000 Claw Poles Alstom

500 20000 3600 20000 Induction ASIRobicon

100 1500 6000 15000 Induction Rotatek Finland

100 730 3600 14000 Induction ABB

100 400 3600 9000 Induction Schorch

40 400 10000 70000 Permanent magnet S2M

50 2000 20000 50000 Permanent magnet Calnetix

20 450 5500 40000 Permanent magnet Reuland Electric

3.7 100 3000 12000 Induction Siemens

1 150 25000 Switched reluctance SR Drives

1 20 15000 Switched reluctance Rocky Mountain Inc.

1.3 Solid-rotor constructions in high-speed induction machines

In the induction motor, in order to produce an electromagnetic torque Tem, and a corresponding

electric output powerPe the rotor mechanical rotating angular speed R must differ from the

rotating synchronous angular speed S of the stator flux. This speed difference guarantees the

induction in the rotor. In fact, the name induction motor is derived from this phenomenon.

Corresponding differences between the rotor electrical angular speed Rand the supply

electrical angular speed S as well as the rotor rotating frequencyfR, and the supply frequency

fS are also present. The differences are usually described with the per-unit slip, which is defined

as:

S

sl

S

RS

S

sl

S

RS

S

sl

S

RS

f

f

f

ff

s =

==

==

=

. (1.1)


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Here, sl describes the mechanical angular slip speed of the rotor, sl the electrical slip angular

speed of the rotor andfsl the electrical slip frequency in the rotor. In motoring mode the slip s is

positive and in generating mode the slip is negative.

The relation between the angular speeds, pole pair numberp, torque and power may be written

as

emR

emR

emRe

2T

p

fT

pTP ===

(1.2)

The slip frequency fsl and the slip angular speed sl in the rotor are of great importance,

especially in solid-rotor machines since the slip angular speed, for instance, has a significant

role in determining the magnetic flux penetration in the rotor. The slip angular speed is one of

the factors determining the torque produced by the rotor. The I2R losses, however, in the rotor

depend on the per-unit slips. For the design of a high-efficiency solid-rotor machine, one of the

design targets should be the minimisation of the per-unit slip.

Solid-rotor induction motors are built with a rotor core made of a solid single piece of

ferromagnetic material. The simplest solid rotor is, in fact, a smooth steel cylinder. The

electromagnetic properties of such a rotor are, however, quite poor, as, e.g., the slip of the rotor

tends to be large, and thus several modifications of the solid rotor may be listed. A common

property of the rotors called solid rotors is the solid core material that, in all cases, forms at least

partly the electric and magnetic circuits of the rotor. The first performance improvement in a

solid rotor is achieved by slitting the cross section of the rotor in such a way that a better flux

penetration into the rotor will be enabled. The second enhancement is achieved by welding

well-conducting non-magnetic short-circuit rings at the end faces of the rotor. The ultimate

enhancement of a solid rotor is achieved by equipping the rotor with a proper squirrel cage. In

all these enhancements the rotor ruggedness is best maintained by welding all the extra parts to

the solid-rotor core. Smooth solid-steel rotors may also be coated by a well-conducting

material. Five different basic variants of solid-rotor constructions are schematically shown in

Fig. 1.2.

The smooth solid rotor is the simplest alternative and thus the easiest and the cheapest to

manufacture. It also has the best mechanical and fluid dynamical properties, but it has the

poorest electrical properties. In practice, the manufacturing of a smooth solid rotor is not


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24

profitable because by milling axial slits into the rotor it is possible to get considerably more

power, a slightly better power factor and a higher efficiency than it may be achieved with a

smooth rotor, and the machining costs remain moderate. Rotor coating, end rings and squirrel-

cage structures raise the manufacturing demands and costs, but these structures boost the motor

torque and properties in a considerable way. For example, according to the experience of the

author, a smooth solid rotor equipped with copper end rings produces twice as much torque at acertain slip as the same rotor without end rings and a motor with a copper-squirrel-cage solid

rotor gives three to four times as much torque as the same motor with a smooth solid rotor. The

fundamental rotor losses in a copper-cage solid rotor are only a fraction of those of a smooth

solid rotor. In addition, a squirrel-cage rotor construction gives a clearly better power factor

comparable to the power factor of a standard induction motor than a smooth rotor one.

The solid-rotor induction motor construction offers several advantages:

High mechanical integrity, rigidity, and durability. The solid rotor is the most stable

and of all rotor types it maintains best its balance.

High thermal durability.

Simple to protect against aggressive chemicals.

High reliability.

Simple construction, easy and cheap to manufacture.

Very easy to scale at large power and speed ranges.

Low level of noise and vibrations (if smooth surface).


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a)

b)

c)

d )

e)

Fig. 1.2. Solid-rotor constructions: a) smooth solid rotor, b) slitted solid rotor, c) slitted solid rotor with

end rings, d) squirrel-cage solid rotor, e) coated smooth solid rotor. Gieras (1995)


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On the other hand, a solid-rotor induction motor has a lower output power, efficiency, and

power factor than a laminated rotor cage induction motor of the same size, which are

disadvantages that are mainly caused by the high and largely inductive impedance of the solid

rotor. The solid rotor impedance and its inductive part can be diminished in one of the

following ways:

1. The solid rotor may be constructed of a ferromagnetic material with the ratio of

magnetic permeability to electric conductivity as small as possible.

2. Using axial slits to improve the magnetic flux penetration to the solid ferromagnetic

rotor material.

3. A layered structure in the radial direction of the rotor may be made of appropriate

ferromagnetic and non-ferromagnetic high-conductivity materials (coated rotor).

4. A layered structure in the axial direction of the rotor may be made of appropriate

ferromagnetic and non-ferromagnetic high-conductivity materials (end-ring structure).

5. Use of a squirrel cage embedded in the solid ferromagnetic rotor core material.

6. The effects of the high impedance may be offset by the use of an optimum control

system.

7. Use the solid rotor in high-speed applications when the per-unit slip is low. The higher

the motor rotating frequency is the less important the rotor impedance will be. For

example: The rotor needs a 2 Hz absolute slip to produce the needed torque. If the

motor rotating frequency is 50 Hz the per-unit slip is 4 %, which means that 4 % of the

air-gap power is lost in the rotor copper losses. If the rotating frequency is 200 Hz the

same absolute slip results in a 1 % per-unit slip and, correspondingly, in a 1% per unit

rotor copper losses.

Solid-rotor induction motors can be used as:

High-speed motors and generators.

Two- or three-phase motors and generators for heavy duty, fluctuating loads,

reversible operating, and so forth.

High-reliability motors and generators operating under conditions of high temperature,

high acceleration, active chemicals, and so on.

Auxiliary motors for starting turbo-alternators.

Flywheel applications.


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27

Integrated machines. The rotating part of the load machinery forms the rotor, for

example conveyer idle, where the stator can be outside or inside of the rotor.

Eddy-current couplings and brakes.

1.4 Objectives of the work

The problem of calculating the magnetic fields in solid rotors has been a subject of intensive

study from the 40s till the 70s. The investigations were carried out with strong relation to the

smooth solid rotor and conventional speeds, and because there were no powerful computers

available, the calculation models were strongly simplified. Most experiments showed that the

electrical properties of the solid-rotor IM are not good enough.

Since the use of high-speed machines became more popular from the beginning of the 1990s a

few FEM studies about solid-rotor IMs have been published, but still the activities remained

low in this specified field.

The present study is done to establish a fast practical method for the design purposes

determined by the manufacturer of solid-rotor motors. The research has seven main objectives.

1) To create an analytical, multi-layer transfer-matrix method (MLTM method) based

calculation procedure for a slitted solid rotor equipped with copper end rings in order to enable

an accurate enough estimation of the behaviour of the electromagnetic fields in the slitted solid

rotor. When the field problem is solved the motor air-gap power is found by integrating the

Poynting vector over the rotor surface. The rotor behaviour is then connected to the traditional

equivalent circuit behaviour of the induction motor. 2) To introduce an analytical procedure by

means of which it is possible to precisely enough determine the losses of the solid-rotor IM. 3)To find the best length to diameter ratio for a copper end ring slitted solid rotor. 4) To find the

best possible practical slitting patterns for the industrial motor solid rotor with copper end rings,

5) to introduce the power-dependent utilization factors for different types of solid rotors based

on the practical research results reached at LUT, 6) to compare the analytically found

electromagnetic results with the Finite Element Method (FEM) based solutions, and 7) acquire a

practical proof for the given theories by making careful measurements with appropriate

prototypes. The output powers of the prototypes vary between 250 kW and 1000 kW as the

speeds of the prototypes vary between 8400 min-1 and 12000 min-1. The main dimensions of the

250 kW 300 kW prototype machines are: a 200 mm air-gap diameter, a 280 mm stator stack

effective length.


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This work strongly focuses on the electromagnetic phenomena of the solid-rotor machine,

irrespective of the fact that mechanic and thermodynamic studies are of essential importance,

especially as high-speed machines are concerned. Usually, in practice, all of these three

demanding scientific fields need their own specialists to solve the exacting challenges in the

different fields. For that reason, the need of limiting this study to the electromagnetic

phenomena should be acceptable.

1.5 Scientific contribution of the work

In summary, the main scientific contributions of the thesis are:

1. The further development of the well-known multi-layer transfer-matrix method to be

used, especially, for the calculation of high-speed slitted solid-rotor induction motors.

Improvement of the multi-layer transfer-matrix method was achieved by introducing

into the method a new end-effect factor and a new curvature factor for slitted solid-

rotors equipped with well-conducting end rings. The new factors are functions of theslit depths.

2. Definition of the best possible practical slitting of solid rotors equipped with well-

conducting end rings for high-speed induction motors in the medium power range.

3. Definition of the best possible rotor active length to diameter ratio for slitted solid-rotor

induction motors with well-conducting end rings.

4. Introducing of the power-dependent utilization factors for different types of solid

rotors.

5. Introduction of a new method to reduce the permeance harmonic content in the air-gap

flux density distribution by means of a new geometrical modification of a semi-

magnetic slot wedge. The slot wedge is formed as a magnetic lens.

Apart from these scientifically new contributions, the thesis also contributes, especially to the

practical engineer, in a valuable way, which may be summarized to be the following:


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1. An analytical electromagnetic and accurate enough - analysis of the solid-rotor

induction machine is introduced. The method is very useful in every-day practical

electrical engineering.

2. Discussion on the analysis of the analytical harmonic power loss calculation in solid

rotors. Methods of minimizing the harmonic power loss in the rotor surface are also

widely discussed.

3. New practical information on selecting the flux densities in the different parts of solid-

rotor induction machines in the medium speed and power range.

4. Some measures of diminishing the time harmonics caused by the frequency converter

are briefly introduced.

Several end-effect factors are presented in the literature on the subject. Usually, these factors

are introduced for a smooth solid rotor. They are based on the calculation of the penetrationdepth, and should thus be a function of the rotor slip frequency. In practice, in a deeply slitted

solid rotor with well-conducting end rings, the axial rotor currents penetrate as deep as the slits

are. And, in practice, this current penetration depth is not depending on the slip when a normal

slip range of not more than a few percents is used. It is thus possible to use the real dimensions

of the end rings in the end-ring impedance calculations. The analysis assumes also that the

inductance of the end ring is negligible compared to the inductance of the slitted part of the

rotor.

Furthermore, a new curvature factor is defined for slitted solid rotors to be used in the MLTM

method when rectangular coordinates are used.

Slitting patterns for solid rotors have been studied earlier, but the examinations were in different

ways restricted; they were not done for high-speed machines, the parameter variation was done

within a very narrow range, the electromagnetically best slitting alternatives could be found but

the practical manufacturing conditions were disregarded.

According to the knowledge of the author, the utilization factors introduced in this thesis for

different types of solid-rotor induction motors have not been presented earlier. However, the

utilization factors for copper-coated solid-rotor induction motors were presented by Gieras

(1995).


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1.6 Outline of the work

The multi-layer transfer-matrix method for a solid rotor was introduced by Greig (1967). Later,

several authors have used this method. The substitute parameters for a slitted solid rotor were

introduced by Freeman (1968). These form the basics for the calculation procedure introduced

here. In the second chapter, the history of the field calculation problem in the solid rotor is

discussed. The MLTM principles are repeated in chapter two.

Loss calculation of the solid-rotor IM is also one of the main objectives. When a solid rotor is

used extra attention must be paid to the eddy currents on the surface of the rotor solid steel.

Eddy currents are caused by the spatial and time harmonics of the air-gap magnetomotive force

(mmf) and the permeance harmonics as well. This is discussed in chapter three.

In chapter four the slitted solid rotor is examined and the MLTM and FEM calculation results

are compared. Also the measured results are given.

The conclusions of the research are given in chapter five.


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2. Solution of the electromagnetic fields in a solid rotor

This chapter describes the development and gives a review of the analytical methods that have

been introduced for the solving of the electromagnetic fields in solid-steel rotors. Since the

conventional induction machine theory proved to be inadequate for solid-rotor machines, the

need has grown to improve the methods of investigation. It has become necessary to determine

the solid-rotor machine performance directly based on the analysis of the electromagnetic

fields. The specific problems such as saturation, the effect of the finite axial length and rotor

curvature also affect the performance of the motor greatly and are, for this reason, of most

significant importance. In this study some of the known methods are combined and further

investigated in order to find a solution, which, in an appropriate way, gives consideration to all

the important rotor phenomena.

Although a smooth solid rotor is an extremely simple construction, the calculation of itsmagnetic and electric fields is a demanding process because the rotor material is magnetically

non-linear and the electromagnetic fields are three-dimensional. Thus, to solve the solid-rotor

magnetic and electric fields fast and accurately enough is a demanding task. In the conventional

laminated squirrel-cage rotor induction motor design the magnetic and electric circuits can be

assumed to be separated from each other in the stator as well as in the rotor so that the electric

circuit flows through the coils and the magnetic circuit flows mainly through the steel parts and

the air-gap of the machine. For this reason, these phenomena can be examined separately.

Furthermore, in a traditional induction motor the magnetic circuit is made of laminated electric

sheets and end rings are included in the squirrel cage, and thus, without losing accuracy, it has

been possible to perform the examination in two dimensions and the non-dominant end effects

could be studied separately. In a solid rotor the steel material forms a path for the magnetic flux

and for the electric current, and, therefore, three-dimensional effects and non-linearity have to

be taken into consideration. Hence, the standard linear methods of analysis in which only

lumped parameters are considered, are no longer valid.

The rotor field solution could be solved by the three-dimensional FEM calculation, but it takes

far too much time to be used in every-day motor design proceeding. Besides, the modelling of a

rotation movement even more complicates the FEM calculation. Therefore, a three-dimensional


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analytical solution for the rotor fields has to be found. The ultimate simplification is to solve the

Maxwells field equations assuming a smooth rotor and a magnetically linear rotor material.

The literature in the field widely deals with the analysis of the solid rotor, especially in the

1950s, 1960s and 1970s. Research was carried out with the objective to maximize the starting

torque and to minimize the starting current and, further, to simplify the rotor construction of an

induction machine.

In the articles it is commonly supposed that the rotor is infinitely long. Another assumption

made is that the rotor material is magnetically linear or the rotor material has an ideal

rectangularBH-curve. The assumption of an infinitely long rotor brings as a result a two-

dimensional analysis, but to achieve a good accuracy the end effects should be taken into

consideration. On the presumption of the rotor material being magnetically linear, a constant

value of 45 is given to the phase angle of the rotor impedance. The constant phase angle is

contrary to many experimental results, which have shown that the phase angle of non-laminated

steel rotors is far less than 45.

An important feature of the solid-rotor induction machine is that the magnetic field strength at

the surface levels of the rotor is usually sufficient enough to drive the rotor steel deep into the

magnetic saturation. The limiting non-linear theory of the flux penetration into the solid-rotor

material considers that the flux density within the material may exist only at a magnitude to a

saturation level. This theory was used by MacLean (1954), McConnell (1955), Agarwal(1959),

Kesavamurthy (1959), Wood (1960d), Angst (1962), Jamieson (1968a), Rajagopalan (1969),

Yee (1972), Liese (1977) and Riepe (1981a). This rectangular approximation to the BH-curve is

good only at very high levels of magnetisation. This analysis gives a constant value of 26.6 tothe rotor impedance phase angle when the applied magnetizing force is assumed to be

sinusoidally distributed (MacLean 1954, Chalmers 1972, Yee 1972). Both the linear theory and

the limiting non-linear theory produce a constant power factor for the rotor impedance

independent of the rotor slip, material and current. That is, however, contrary to the

experimental results. In practice, the phase angle of the rotor impedance is somewhere between

these two extremes given by the linear theory and the limiting non-linear theory. Usually,

magnetic material saturation is a disadvantage that complicates the phenomena and decreases

the performance. It could, however, be determined that the saturation effects of the solid-rotor

steel, in this particular case, are beneficial since they increase the solid-rotor power factor. The

equivalent circuit approach was used by McConnell (1953), Wood (1960a), Angst (1962),


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Dorairaj (1967b), Freeman (1968), Sarma (1972), Chalmers (1984), and Sharma (1996). Cullen

(1958) used the concept of wave impedance.

To define the impedance of the solid rotor a non-linear function for theBH-curve must be used.

The non-linear variation of the fundamental B1-H curve is included in its entirety by

substituting the equation B1=cH(1-2/n), where c and n are constants. This fits the magnetisation

curve well. This form was used by Pillai (1969). He concluded that the rotor impedance phase

angle varies according to the exponent ofH, lying between 35.3 and 45, while n varies

between 2 and , respectively. Test results showed that the real phase angle of the rotor

impedance approaches Pillais value when the slip increases and the magnetic field strength

drives the surface of the rotor steel into the magnetic saturation. Respectively, at very low slips

the phase angle approaches 45. Thus, the varying range of the phase angle is restricted between

35.3 and 45.

Pipes (1956) introduced a mathematical technique the transfer-matrix technique for

determining the magnetic and electric field strengths and the current density in plane

conducting metal plates of constant permeability produced by an external impressed alternating

magnetic field. This method was later generalised by Greig (1967). Greig calculated the

electromagnetic travelling fields in electric machines. The generalised structure comprises a

number of laminar regions of infinite extent in the plane of lamination and of arbitrary

thickness. The travelling field is produced by an applied current sheet at the interface between

two layers. It is distributed sinusoidally along the plane of the lamination and flowing normally

to the direction of the motion. The transfer matrix calculates the magnetic and electric field

strengths of the following plane from the values of the previous plane using prevailing material

constants. The method is called multi-layer transfer-matrix method (MLTM method).

The MLTM method divides the rotor into a large number of regions of infinite extent. The

original MLTM method does not consider the rotor curvature, material non-isotropy or the end

effects, but the method gives consideration to the non-linearity of the material, because the

permeability and the conductivity of the rotor material are presumed to be constants in each and

every region separately. The tangential magnetic field strength and the normal magnetic flux

density will be calculated in every region boundary using the suppositions mentioned earlier.

After that the permeability and the conductivity in each region have been defined and hundreds

of regions have been calculated, it is possible to achieve very accurate results. (Pyrhnen

1991a).


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The method described above was later developed by Freeman (1970) who published a new

version on the technique used for polar coordinates. This technique was also used by Riepe

(1981b). Yamada (1970), Chalmers (1982) and Bergmann (1982) used the MLTM method in

the Cartesian coordinates.

2.1 Solution of the electromagnetic rotor fields under constantpermeability

In the following analysis, a field solution is derived for a linearized, smooth rotor of finite

length. The solution is written in the form of a Fourier-series. This method was first used by

Bondi (1957) and later developed by Yee (1971). The linear method requires solving of

Maxwells equations. The field solutions are approximate, because the solution in closed form

becomes impossible without some simplifications. These hypotheses are:

The rotor material is assumed to be linear so its relative permeability and conductivity are

constants. The material is homogenous and isotropic. There is no hysteresis.

The surface of the rotor is smooth.

The curvature of the rotor is ignored. The rotor and stator are expanded into flat, infinitely

thick bodies. Equations are written in rectangular coordinates.

The stator permeability is infinite in the direction of the laminations.

The stator windings and currents create an infinitesimally thin sinusoidal current sheet on

the surface of the stator bore. This current sheet does not vary axially.

The magnetic flux density normal to the end faces is zero.

The radial magnetic flux density in the air-gap does not vary in the radial direction. The

mistake made here is negligible when the air-gap is small compared to the diameter of the

rotor.

In the applied method a coordinate system fixed with the rotor is used, as it is shown in Fig. 2.1.

The origin is at the surface of the rotor and axially at its midpoint. The z-axis is taken in the

axial direction. The y-axis is normal to the rotor surface and the x-axis is in the tangential

direction, i.e. it is in circumferential direction. When the rotor is rotating at a slip s in the

direction of the negativex-axis, its position in the stator coordinates can be written as

p

rtsxx

s

RS )1( = , (2.1)


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wherep is number of pole pairs, ris rotor radius, tis time and s is stator angular speed.

y

xz

Fig. 2.1. Coordinate system at the surface of the rotor.

The next abbreviation is taken into use. The constant a is dependent on the dimensions of themachine

p

a

= , where p is pole pitch,

p

Dp 2

= . (2.2)

Equation (2.1) can be rewritten now

tsaxtax sR

sS +=+ (2.3)

Henceforth, the superscript R, which indicates to coordinate fixed to the rotor, will be left out.

The differential forms of Maxwells equations have to be used as a starting point. Amperes law

relates the magnetic field strength Hwith the electrical current density Jand the electric flux

density D. Faradays induction law determines the connection between the electric field

strength E and the magnetic flux density B. Gauss equations definitely reveal that the

divergence ofB is zero and the divergence ofD is charge density, i.e.B has no source andD

has the source and the drain.

t

DJH += , ( 2.4)


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t

BE = , (2.5)

0= B , (2.6)

= D , (2.7)

The latter part of equation (2.4) representing Maxwells displacement current is omitted,

because the problem is assumed to be quasi-static, i.e. Maxwells displacement current is

negligible compared with the conducting current at frequencies which are studied in solid-rotor

materials, see App. C.

In addition, the material equations are needed:

ED = , (2.8)

HB = , (2.9)

EJ = , (2.10)

where is the material permittivity, is the permeability of the material and its conductivity.

A two-dimensional eddy-current problem can be formulated in terms of the magnetic vector

potentialA, from which all other field variables of interest can be derived. The magnetic vector

potential is defined as a vector such that the magnetic flux densityB is its curl:

BA = . (2.11)

Equation (2.11) does not define the magnetic vector potential explicitly. Because he curl of the

gradient of any function is equal to zero, any arbitrary gradient of a scalar function can be added

to the magnetic vector potential while equation (2.11) is still correct. In case of static and quasi-

static field problems the uniqueness of equation (2.11) is ensured by using the Coulomb gauge,

stating the divergence of the magnetic vector potential to be zero everywhere in the space

studied


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0= A . (2.12)

When equation (2.11) is substituted to Faradays law equation (2.5) we get

0=

+ At

E . (2.13)

The sentence in parenthesis has no curl and may thus be written as a gradient of a scalar

function . Now, the electric field strength can be written in the following form

=

t

AE . (2.14)

The charge density can be assumed to be negligible in well-conducting solid-rotor material.

Therefore, the divergence of the electric field strength is zero.The reduced scalar potential

describes the non-rotational part of the electric field strength. The non-rotational part is due toelectric charges and polarisation of dielectric materials. However, in a two-dimensional eddy-

current problem the reduced scalar potential must equal zero, see App. D.

Using equations (2.9), (2.10), (2.11) and (2.14) and keeping permeability and conductivity

as constants, equation (2.4) can be written

t

==A

AAA 2)()( . (2.15)

When the Cartesian coordinates are used and the Coulomb gauge, equation (2.12), is valid, the

differential equation ofA can be expressed by

t

A

z

A

y

A

x

A iiii

=++

2

2

2

2

2

2

, (2.16)

where i isx,y, orz(Yee 1971).

Because all fields in the induction machine may be assumed to vary sinusoidally as a function

of time, a steady state time-harmonic solution may be found in the analysis. The vector

potentialA is considered. It can be expressed in a time-harmonic form by


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[ ]tszyxtzyx sje),,(Re),,,( AA = , (2.17)

where A is a complex and only position dependent vector. The space structure of the stator

winding of the induction machine causes the vector potentialA to vary in the direction of thex-

axis both as a function of place x with the term ejax and as a function of time twith the term

ej ss t . The vector potential is obtained in form of a complex vector function

[ ])(j se),(Re),,,( tsaxzytzyx += AA . (2.18)

Now, equation (2.16) can be written as a complex exponent function

iii Aa

z

A

y

A)( 22

2

2

2

2

+=+ , (2.19)

wherep

s

j2j

ds == , (2.20)

dp is the penetration depth and describes the wave penetration to a medium. The equations

(2.16) - (2.19) can be written analytically as phasor equations. For instance, equation (2.4) in a

time harmonic form is

DJH j+= . (2.21)

Using the annotation ,which describes the variation of the fields in the axial direction, and for the air-gap length we get

r

2

+= a . (2.22)

Pyrhnen(1991a) repeated a mathematical deduction to the solution, which is convergent to the

solution given by Yee (1971). In deriving the solution for the rotor fields the necessary

boundary conditions to the solution are chosen in a convenient manner as:

1. The current has no axial component at the ends of the rotor.


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2. The magnetic flux density has no axial component at the ends of the rotor.

3. All field quantities disappear, when y approaches -, because the flux penetrates into the

conducting material and attenuates.

4. The machine is symmetrical inxy-plane.

In addition, the depth of the penetration is assumed to be much smaller than the pole pitch.

The simplified equations in closed form for the vector potential in the x, y and z-direction are:

(Pyrhnen 1991a)

)(j se)

2sinh(

)sinh()ee(

)2

sinh(

)sinh(e tsaxyayy

x L

z

L

zGA

+

+= , (2.23)

)(j se

)2

sinh(

)sinh()ee(j tsaxayyy

L

zGA

+= , (2.24)

)(j se)

2sinh(

)cosh()

2coth()

2coth(ej tsaxyz L

zaLaLGA

+

+= , (2.25)

where

++

=

)2

coth()2

coth()(

j

r

2

00S

LaLa

KIG

, Na

p

mK

0 = . (2.26)

In the rotor the magnetic flux density equations are:

( ) )(j se)

2sinh(

)cosh(ee

)2

sinh(

)cosh()

2coth()

2coth(ej tsaxyayyx L

z

L

zaLaLGB

+

+

+= ,

(2.27)


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40

( ) )(j se)

2sinh(

)cosh(ee

)2

sinh(

)cosh()

2coth()

2coth(e tsaxyayyy L

z

aL

za

a

LaLGaB

+

+

++= ,

(2.28)

)(j se)

2sinh(

)sinh(

)2

sinh(

)sinh(e tsaxyz L

z

L

zGB

+

= . (2.29)

The tangential and the axial magnetic flux components per unit width on the surface of the rotor

are found by integrating the respective flux densities:

)(j0

se)

2

sinh(

)cosh(1

)

2

sinh(

)cosh()

2coth()

2coth(jd tsax

xx L

z

aL

zaLaLGyB

+

++== , (2.30)

)(j0

se)

2sinh(

)sinh(

)2

sinh(

)sinh(d tsaxzz L

z

L

zGyB

+

== . (2.31)

The preceding field equations with respect to z are shown graphically in Fig. 2.2. As it is

illustrated in the figure,Az andHz are not zeros at the ends of the rotor, as it was required by the

boundary conditions. This is a result of the approximations made to obtain the solutions. The

dotted line sketches the forms of the actual distributions.


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AZ

Ax

HZ

Z

Hx

x

1

0

L / 2

L / 2

L / 2

1

0

1

0

Fig. 2.2. The axial distribution of the rotor fields at the surface of the rotor at standstill. The quantities

are normalized with respect to theAz,Hx and x values atz= 0 (Yee 1971). a) Magnetic vector

potential aty = 0, b) magnetic field strength aty = 0, c) magnetic flux per-unit length.

2.2 Calculation of a saturated solid-rotor

The electromagnetic fields in saturated rotor material can be solved with the MLTM method,

where the rotor is divided into regions of infinite extent. Fig. 2.3 describes the multi-layer

model and the coordinates used, Greig (1967).

In general, the current sheet

{ })(j se'Re taxJJ += , (2.32)

lies between any two layers. Regions 1Kare layers made of material with resistivity k and

relative permeability k. The problem is to determine the field distribution in all regions, and

hence, if required, the power loss in and forces acting on any region.


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K B

H

K K K-1

K-1

K-1 B

H

K-1 K-1 K-2

K-2

B

Hk+1k+1

B

H

k

kk B

H

k k k-1

k-1 k-1 k-1

y

x

z

H -J'k

3B

H

3 3

2

22 B

H

2 2 1

11 1 1

.

..

..

y = gK-1

y = g1

y = g2

k-1

k+1k+1

k+1

.

Fig. 2.3. Original two-dimensional multi-layer model (Greig 1967).

A stationary reference frame is chosen in which the exciting field travels with velocity s/a. A

region k, in which the slip angular speed is k = sks, is therefore travelling at velocity (1-sk)s/a relative to the stationary reference frame (Greig 1967). Please note that in all the rotor

regions the slipskis the same and a constant. In the stator regions the slip is zero.

Consider a general region kof thickness dk, as it is given in Fig. 2.4. The normal component of

the flux density on the lower boundary is By,k-1, and the tangential component of the magnetic

field strength is Hx,k-1. The corresponding values on the upper boundary are By,k and Hx,k,

respectively (Greig 1967).

It is assumed that the regions may be considered planar, all end effects are neglected, as it has

been done for the magnetic saturation too; also the displacement currents in the conducting


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43

medium are considered to be negligible. The current sheet varies sinusoidally in the x direction

and with time; it is of infinite extent in thex direction, and of finite thickness in they direction.

Maxwells equations may be solved when the boundary conditions are as follows: (Greig 1967)

1. By is continuous across a boundary.

2. All field components disappear aty = .

3. If a current sheet exists between two regions, then '1 JHH kk = .

region k + 1 B

H

k+1 k+1 k

kregion k B

H

k k k-1

k-1

y = gk

region k - 1y = gk-1

dk

k

Fig. 2.4. Definition of the properties and dimension of region k(Freeman 1968).

The following matrix equation may be written for region k, according to Greig (1967):

[ ]

=

=

1,

1,

1,

1,

,

,

)cosh()sinh(

)sinh(1

)cosh(

kx

ky

k

kx

ky

kkkkk

kk

k

kk

kx

ky

H

B

H

B

dd

dd

H

BT

, (2.33)

wherek

k

k a

0j= and kkkk sa 0s

2 j+= (2.34)

and [Tk], following Pipes (1956), is the transfer matrix for the region k. In the top region on the

boundarygK

1,1, = KyKKx BH . (2.35)

In the top region K the magnetic flux density and the magnetic field strength have to vanish

gradually to zero according to boundary condition (2), thus (Greig 1967)

)(1,,

1e ygKyKyKKBB

= , (2.36)


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44

)(1,,1e ygKxKKx

KKHH

= . (2.37)

Solving the field in the bottom region on the boundaryg1

1,11, yxBH = . (2.38)

In the region 1 the magnetic flux density and magnetic field strength must approach zero as y

diminishes, it can be written (Greig 1967)

)(1,1,

11e gyyy BB= , (2.39)

)(1,1,

11e gyxx HH= . (2.40)

The transfer matrix can be used as follows, considering the boundary conditions (1) and (3).

The current sheet lies between regions kand (k+1). (Greig 1967).

[ ][ ] [ ]

=

1,

1,21

,

,

x

y

kk

kx

ky

H

B

H

BTTT L , (2.41)

[ ][ ] [ ]

=

+

',

,121

1,

1,

JH

B

H

B

kx

ky

kKK

Kx

KyTTT L . (2.42)

The analysis above may be programmed to compute the electromagnetic fields and power flow

at all boundaries. The computing can be initiated by using a presumed low value of thetangential field strength Hx,1 at the inner rotor boundary. The transfer matrix technique then

evaluatesBy,k andHx,k at all inter-layer boundaries up to the surface of the rotor. At this interface

Hx,k corresponds to the total rotor current. This rotor model may be combined with a

conventional equivalent circuit representation of the air-gap and the stator. Iterative adjustment

ofHx,1 is made to adapt the conditions at the rotor surface.

AsBy,k andHx,k are resolved at all inter-region boundaries, it is then a simple matter to calculate

the power entering a region. The Poynting vector in the complex plane is

.* ,, kxkzk HES = (2.43)


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The time-average power density in (W/m2) passing through a surface downwards at gk may be

found by using the following expression: (Freeman 1968)

{ }* ,,,in Re5.0 kxkzk HEP = , where k= 1, 2, ..K. (2.44)

Ez,k is the component of the electric field strength in thez-direction and it may be written as:

kyk

kz Ba

E ,,

= . (2.45)

The net power density in a region is the difference between the power density in and the power

density out (Greig 1967):

( )

=

*1,1,

*,,

s

2Re kxkykxkyk HBHB

aP

. (2.46)

The mechanical power density evolved by the region under slipsk is (Greig 1967)

)1(.mech kkk sPP = . (2.47)

The ohmic lossI2R elaborated by the region is (Greig 1967)

kkkk PsPP = ,mech . (2.48)

2.2.1 Definition of the fundamental permeability in a non-linear material

In a saturable material sinusoidally varying magnetic field strength creates a non-sinusoidal

magnetic flux density (Bergmann 1982). The amplitude spectrum of this flux density can be

numerically defined with the DC-magnetisation curve of the material. Fig. 2.5 shows how the

flattenedB(t)-wave contains a fundamental amplitude which is considerably higher than the

real maximum value. The harmonics may be ignored in the analysis of the active power

because, according to the Poynting vector, only waves with the same frequency create power.

So, the saturation dependent fundamental permeability of the material has to be defined. The

fundamental amplitude 1B of the Fourier series of the flux density is obtained by a numerical

integration:


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46

=

0

1 )(d)sin()(

2 tttBB . (2.49)

The fundamental permeability of a particular working point is defined as

HBH

)( 11 = . (2.50)

B

H

H

t

H(t)

B (t)

B (t)

1

B1

t

B

H

Fig. 2.5. The definition of the fundamental magnetic flux density B1(t) produced by an external

impressed sinusoidally alternating magnetic field strengthH(t) and theB1-Hcurve with DC-

magnetizing curve.

2.2.2 Rotor impedance

The rotor fundamental magnetomotive force in the air-gap, referred to the stator, is

a

HxHI

p

Nm xaxx

pj

2de'

2

42 R

0j

RR1 ===

, (2.51)


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47

from which the rotor current referred to the stator is found:

xH

Nam

pI RR

2

j'

= . (2.52)

The air-gap flux of the machine is obtained by integrating the radial flux density at the rotor

surface over a pole pitch. Faradays induction law gives an equation for the rotor voltage per

phase referred to the stator:

y

ax

y Ba

LNxLB

NU

p

p

Rs

2

2-

jRsR

2

2jde

2j'

== . (2.53)

Finally, the rotor impedance referred to the stator is found:

x

y

H

B

p

mLN

I

UZ

R

R2

s

R

RR

)(2

'

''

== . (2.54)

2.3 Effects of axial slits in a solid rotor

The performance of an induction machine with a solid-steel rotor can be considerably improved

by slitting the rotor axially. The presence of slits has a significant influence on the eddy current

distribution in the rotor; the slits usher the eddy currents to favourable paths as the torque is

considered. The non-isotropy of the rotor body resulting from the slitting is in contradiction

with the boundary condition of the MLTM method. Thus, the analysis of the rotor fields is now

essentially a three-dimensional problem the solving of which, as the slitted nature of the rotor

surface is to be taken into account, is an extremely complex and laborious task. Slitted rotor

fields were studied by Dorairaj (1967a), Freeman (1968), Jamieson (1968b), Rajagopalan

(1969), Yamada(1970), Bergmann (1982), Jinning (1987) and Zaim (1999).

Jinning (1987) studied optimal rotor slitting. According to his calculation results, the optimal

number of slits is between 5 and 15 per pole pair. The optimal depth of a slit equals

approximately the magnetic flux penetration depth and the ratio between the slit width and the

slit pitch is between 0.05 and 0.15. Zaim (1999) analysed a slitted solid-rotor induction motor

by means of a FEM program, but only a few rotor slit parameters are used. Also Laporte (1994)


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48

investigated optimal rotor slitting, but his treatment of the subject is not expansive enough

either.

A slitted rotor may be solved by means of the MLTM method using substitute parameters for

the permeability and the conductivity of the rotor material in the slitted region. The substitute

parameters are obtained using a slit pitch u, a slit and a tooth width wu and wt, relativepermeability of the tooth t and both slit and tooth resistivityu jat, Fig. 2.6 (Freeman 1968).

Here, it is assumed that the slit is not of a magnetic medium, i.e. u = 1. The method considers

the slitted rotor region to be replaceable by an equivalent homogenous but anisotropic medium.

This assumption, however, leads to a solution, where the field distribution in slits and teeth

regions would be equal. This, in fact, is far from reality, and thus the assumption should be

considered carefully. If the slit geometry becomes more complicated, compared to the

rectangular shapes, or if the wavelength of the travelling wave is small compared to the slit

pitch, the assumption may break down. Possible skewing may not be taken into consideration.

The substitute parameters are:

u

u

u

tt

wwy += , (2.55)

tut

ut

wwx +

= , (2.56)

tuut

utu

ww

+= . (2.57)

wt wu

u

y

x

z

Fig. 2.6. Slitted solid-rotor surface.


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49

2.4 End effects of the finite length solid rotor

In the previous study the rotor was presumed to have an infinite length. Now, the effects of the

finite length are considered.

The problem of the end effects in solid rotors causes an indisputable difficulty. Several of the

authors earlier mentioned did not take these effects into consideration at all. Omitting the

problem may be justifiable if the rotor is equipped with thick end rings which have very low

impedance and which make the current paths nearly axial. However, this supposition is not

valid even in solid rotors with copper end rings because according to the experience of the

author, when a solid rotor with copper end rings is used and the end effects are not considered,

the calculated results give a 10 - 30 percent better torque at the given slip compared to

measured results. Kesavamurthy (1959) introduced an empirical factor to modify the value of

the rotor conductivity to incorporate the correction for the end effects. The author does not

explain how the empirical factor for the end effect correction is achieved. Russel (1958)

assumed that the rotor current density is confined in a thin shell around the rotor. Also

Rajagopalan (1969) used this assumption. Jamieson (1968a) introduced the analysis in which

the eddy currents are assumed to continue in the body of the rotor. He gives an equation for a

correction factor of the end effects. Wood (1960c) made in his analysis a certain approximation,

the validity of which is questioned. Angst (1962) proposed a complex factor that is applicable to

the effective rotor impedance. Deriving the factor involves the solution of the three-dimensional

field problem under constant permeability. Yee (1971), too, solves the three-dimensional field

problem under constant permeab

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7373941 010 High Speed Solid Rotor Induction Motor (1)

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