1.
Lappeenrannan teknillinen yliopisto
Lappeenranta University of Technology
Jussi Huppunen
HIGHSPEED SOLIDROTOR 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 3rd
of
December, 2004, at noon.
Acta Universitatis
Lappeenrantaensis
197
2.
ISBN 9517649819
ISBN 9517649444 (PDF)
ISSN 14564491
Lappeenrannan teknillinen yliopisto
Digipaino 2004
3.
ABSTRACT
Jussi Huppunen
HighSpeed SolidRotor Induction Machine – Electromagnetic Calculation and Design
Lappeenranta 2004
168 p.
Acta Universitatis Lappeenrantaensis 197
Diss. Lappeenranta University of Technology
ISBN 9517649819, ISBN 9517649444 (PDF), ISSN 14564491.
Within the latest decade highspeed 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 highspeed
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 highspeed 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 solidrotor
induction machine. The slip of the solid rotor is reduced notably if the solid rotor is axially
slitted. 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 multilayer transfermatrix 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
4.
to verify the results. Utilization factor data are given for several other prototypes the largest of
which delivers 1000 kW at 12000 min1
.
Keywords: highspeed induction machine, solid rotor, multilayer transfermatrix, harmonic
losses.
UDC 621.313.333 : 621.3.043.3
5.
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 HighSpeed 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 Pyrhönen, 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 Niemelä for his valuable comments. I also thank the laboratory
personnel Jouni Ryhänen, 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 challenging
working 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 preexaminers Professor Antero Arkkio, Helsinki University of Technology,
and Dr. Jouni Ikäheimo, ABB Motors.
Financial support by the Imatran Voima Foundation, Finnish Cultural Foundation, South
Carelia regional Fund, Association of Electrical Engineers in Finland, Walter Ahlström
Foundation, Jenni and Antti Wihuri Foundation, Teknologiasta Tuotteiksi Foundation and The
Graduate School of Electrical Engineering is greatly acknowledged.
Most of all, to Maiju, Samuli and Julius: Your simple child’s 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
6.
Contents
ABBREVIATIONS AND SYMBOLS.........................................................................................9
1. INTRODUCTION...............................................................................................................15
1.1 APPLICATIONS OF HIGHSPEED MACHINES.....................................................................18
1.2 HIGHSPEED MACHINES..................................................................................................20
1.3 SOLIDROTOR CONSTRUCTIONS IN HIGHSPEED 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 PERMEABILITY
34
2.2 CALCULATION OF A SATURATED SOLIDROTOR.............................................................41
2.2.1 Definition of the fundamental permeability in a nonlinear material ..................45
2.2.2 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 highconductivity 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 SOLIDROTOR 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 airgap 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 highspeed machines............................................................94
3.4 RESISTIVE LOSSES OF THE STATOR WINDING .................................................................94
3.5 LOSS DISTRIBUTION AND OPTIMAL FLUX DENSITY IN A SOLIDROTOR HIGHSPEED
MACHINE ........................................................................................................................96
3.6 RECAPITULATION OF THIS CHAPTER ..............................................................................97
7.
4. ELECTROMAGNETIC DESIGN OF A SOLIDROTOR INDUCTION MOTOR ..99
4.1 MAIN DIMENSIONS OF A SOLIDROTOR INDUCTION MOTOR ...........................................99
4.1.1 Utilization factor....................................................................................................99
4.1.2 Selection of the L/Dratio....................................................................................103
4.1.3 Slitted rotor with copper end rings......................................................................104
4.1.4 Effects of the endring dimensions .....................................................................108
4.2 DESIGN OF SLIT DIMENSIONS OF A SOLID ROTOR .........................................................109
4.2.1 Solving the magnetic fields of a solidrotor induction motor by means of the
FEManalysis.......................................................................................................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
8.
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 crosssection 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 turnend
f frequency
F function
g boundary of region
G 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
9.
10
KC curvature factor
Ker endeffect 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 slits
QS number of stator slots
r rotor radius
r rotor radius vector
R resistance
Rea Reynolds number of axial flow
Rer tip Reynolds number
Reδ 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 layer k
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
10.
11
x function
x, y, z coordinates
X reactance
Yk complex function of layer k
Z impedance
Greek letters
α factor, endeffect factor, angle
β complex function
βδ flux distortion factor
γ factor
γ complex function, a measure of field variation in the axial direction
δ airgap length
ε temperature coefficient of resistivity, permittivity
ζ function
θ angle
Θ magnetomotive force (mmf)
Λ magnetic conductance
λ complex function of slip associated with penetration depth
µ permeability, dynamic viscosity of the fluid
µ0 permeability of vacuum
µr relative permeability
η efficiency, packing factor
ξ winding factor
ρ resistivity, charge density, mass density of the fluid, material density
σ conductivity, material loss per weight
σ Maxwell's stress tensor
σδ leakage factor
τ lamination thickness
τp pole pitch
τu slot pitch
Φ magnetic flux
χ chord factor
ωs stator angular frequency
Ω mechanical rotating angular speed
11.
12
Subscripts
ave average
c cylindrical shell region, conductor
C Carter
Cu copper
class classical
dyn 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 stator
sl slip
sw switching
t tooth
tot total
u slot, slit
v harmonic of order v
x, y, z coordinates
δ airgap
0 basic value, initial value
1 fundamental, bottom layer
12.
13
Superscripts
R rotor
S stator
Other notations
a magnitude of a
a complex form of a
a vector a (in x, y, z coordinates)
a complex form of vector a (timeharmonic presentation)
aˆ peak value of a
Acronyms
AC alternating current
emf electromotive force
DC direct current
FEM finite element method
IGBT insulated gate bipolar transistor
IM induction machine
MLTM multilayer transfermatrix
mmf magnetomotive force
PMSM permanent magnet synchronous machine
PWM pulse width modulation
SM synchronous machine
14.
15
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 “highspeed”. 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 twopole, 50 Hz
machine which has a rotor diameter of 0.96 m. An electrical engineer may not regard a 50 Hz
machine as a highspeed machine. From the motor manufacturer’s point of view a twopole
machine the supply frequency of which is considerably higher than the usual 50 Hz or 60 Hz is
normally considered to be a highspeed machine. However, some motor manufacturers have
called large 3600 min1
machines highspeed machines. The difference of terms used in the
subject can be explained from the other viewpoint, which is that of the power electronics.
Presentday 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 presentday highpower IGBT
technology the switching frequency is limited typically to 1.5 … 6 kHz. Lähteenmäki (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 presentday
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
highspeed machinery. For example, Aglen (2003) reported the application of an 80000 min1
rotating permanent magnet generator to a microturbine and Spooner (2004) described the
project the objective of which was the design of a 6 kW, 120000 min1
axial flux induction
machine to be applied to a turbo charger. This thesis, however, focuses on electric machines
15.
16
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 highspeed 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 loadmachinery 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 solidrotor machine is
needed.
The technology research in the field of highspeed machines has been particularly active in
Finland. Pyrhönen (1991a) studied ferromagnetic core materials in smooth solid rotors.
Lähteenmäki (2002) researched rotor designs and voltage sources suitable for highspeed
machines. His study focused on the design of squirrel cage and coated solid rotors. Saari (1998)
studied thermal analysis of highspeed induction machines and Kuosa (2003) analysed the air
gap friction in highspeed machines. Antila (1998) and Lantto (1999) studied active magnetic
bearings used in highspeed induction machines. However, all of the abovementioned 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, 4pole 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,
6pole 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 solidrotor induction motor. Shalaby (1971) compared harmonic torques
produced by a 3.6 kW, 50 Hz, 4pole induction machine with a laminated squirrelcage 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.
16.
17
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
highspeed solidrotor induction motors. During the latest years research has been focused on
the improving of the efficiency of the highspeed solidrotor 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 highspeed steel rotor runs at quite a low
perunit slip, this indicates that it is possible to reach a good efficiency if the stator losses and
the harmonic content on the airgap flux and the rotor losses are kept low. Research has given
good results and the efficiencies of the highspeed motors have increased up to the level of the
efficiencies of typical 3000 min1
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
(Pyrhönen 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 (Pyrhönen 1993). After the promising
research results, 16 kW, 225 Hz induction motor structures with a smooth, a slitted and a
squirrelcage solid rotor were tested for milling machine applications (Pyrhönen 1996). Later, 8
kW, 300 Hz and 12 kW, 225 Hz copper squirrelcage solidrotor 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 solidrotor 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 cooperation with Rotatek
Finland Oy and LUT.
LUT has also cooperated in the developing of some permanent magnet highspeed machines.
Permanent magnet machines with output powers and rotational speeds of 20 kW, 24000 min1
and 400 kW, 12000 min1
(Pyrhönen 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.
17.
18
Contrarily to this, the simple, rugged solidrotor highspeed 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 ampereturns 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 highspeed 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 highspeed range. Thus, when the motor speed is near
10000 min1
, 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 min1
. This is valid for open motor constructions. If a totally closed construction is used
the benefit of the reduced motor size is lost.
Solidrotor 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 solidrotor shaft and elevated speed is used, the solidrotor 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 highspeed machines
Highspeed solidrotor 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
18.
19
from ca. 50 000 min1
to 10 000 min1
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 FeColamination as well as with CuCrZralloy bars. Also the upper
speed limit for the solidrotor technology is set by the mechanical restrictions and is 100 000
min1
to 20 000 min1
, respectively. But, these mechanical restrictions define the maximal speed
for a certain rotor volume. The limiting power, however, is always defined by the thermal
design 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).
Highspeed 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 highspeed machines gearboxes and couplings can be avoided. The biggest potential
for highspeed machines lies on the field of turbomachinery. 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 speedincreasing gearbox. Such machinery is manufactured by
Atlas Copco, DresserRand, 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.
19.
20
Until the mid1980’s, 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 highspeed installations before
1990. Since the mid1980’s, reliable electric adjustable speed control has been available for
medium voltage, megawattrange, induction motors. As the acceptance of the induction motor
control technology in industry increased, it was only consequent that this technology was
considered to be applied also to highspeed use (Rama 1997).
1.2 Highspeed machines
There are mainly two types of highspeed machines on the presentday market: Highspeed
induction machines and highspeed synchronous machines with permanent magnet excitation.
However, minor research of clawpole synchronous, synchronous reluctance and switched
reluctance highspeed 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 squirrelcage 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, solidrotor 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 nonferromagnetic steel alloy material.
The issue of the stateoftheart highspeed 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 highspeed electric machines that were selected from the
result of a literature search and table 1.2 gives some highspeed electric machine manufacturers.
The trend seems to be that for highspeed motors with power larger than 100 kW the induction
20.
21
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 highspeed
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 min1
are rarely used.
Table 1.1. Some highspeed electric machines selected from literature.
Power/kW Speed/
min1
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 Solidrotor IM, caged Ahrens (2002), prototype
9660 8000 Induction motor Rama (1997), gas compressor
9000 5600 Synchronous motor Khan (1989), feed pump
6900 14700 Laminatedrotor IM McBride (2000), gas compressor
6000 10000 Laminatedrotor IM Gilon (1991), gas compressor
5220 5500 Solidrotor IM, caged LaGrone (1992), gas compressor
2610 11000 Solidrotor IM, caged Wood (1997), compressor
2300 15600 Solidrotor 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 Laminatedrotor IM Joksimovic (2004), compressor
250 8400 Solidrotor IM, end
rings
Huppunen (1998a), blower
200 12000 Solidrotor IM, caged Ikeda (1990), prototype
131 70000 Permanent magnet SM Bae (2003), microturbine
110 70000 Permanent magnet SM Aglen (2003), microturbine
65 30500 Coated
Solid rotor IM, caged
Laminatedrotor IM
Lähteenmäki (2002), prototypes
62 100000 Coated solidrotor IM Jokinen (1997), prototype
60 60000 Coated solidrotor IM Lähteenmäki (2002), prototype
45 92500 Induction Motor Mekhiche (1999), turbocharger
40 40000 Permanent magnet SM Binder (2004), prototype
30 24000 Permanent magnet SM Lu (2000), prototype
21.
22
22 47000 Permanent magnet SM Mekhiche (1999), air condition
21 47000 Laminated rotor IM Soong (2000), cooling compressor
18
12
13500
13500
Solidrotor IM, caged
Solidrotor IM
Solid, slittedrotor IM
Pyrhönen (1996), milling machine
11 56500 Laminated Kim (2001), compressor
Table 1.2. Highspeed standalone electric motor manufacturers in the power range over 100 kW.
Power range/kW Speed range/ min1
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 Solidrotor constructions in highspeed induction machines
In the induction motor, in order to produce an electromagnetic torque Tem, and a corresponding
electric output power Pe the rotor mechanical rotating angular speed ΩR must differ from the
rotating synchronous angular speed ΩS 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 ωR and the supply
electrical angular speed ωS as well as the rotor rotating frequency fR, and the supply frequency
fS are also present. The differences are usually described with the perunit slip, which is defined
as:
S
sl
S
RS
S
sl
S
RS
S
sl
S
RS
f
f
f
ff
Ω
Ω
Ω
ΩΩ
s =
−
==
−
==
−
=
ω
ω
ω
ωω
. (1.1)
22.
23
Here, Ωsl describes the mechanical angular slip speed of the rotor, ωsl the electrical slip angular
speed of the rotor and fsl 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 number p, torque and power may be written
as
em
R
em
R
emRe
π2
T
p
f
T
p
TΩP ===
ω
(1.2)
The slip frequency fsl and the slip angular speed ωsl in the rotor are of great importance,
especially in solidrotor 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 I2
R losses, however, in the rotor
depend on the perunit slip s. For the design of a highefficiency solidrotor machine, one of the
design targets should be the minimisation of the perunit slip.
Solidrotor 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
wellconducting nonmagnetic shortcircuit 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 solidrotor core. Smooth solidsteel rotors may also be coated by a wellconducting
material. Five different basic variants of solidrotor 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
23.
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 a
certain slip as the same rotor without end rings and a motor with a coppersquirrelcage 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 coppercage solid rotor are only a fraction of those of a smooth
solid rotor. In addition, a squirrelcage rotor construction gives a clearly better power factor –
comparable to the power factor of a standard induction motor – than a smooth rotor one.
The solidrotor 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).
24.
25
a)
b)
c)
d)
e)
Fig. 1.2. Solidrotor constructions: a) smooth solid rotor, b) slitted solid rotor, c) slitted solid rotor with
end rings, d) squirrelcage solid rotor, e) coated smooth solid rotor. Gieras (1995)
25.
26
On the other hand, a solidrotor 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 nonferromagnetic highconductivity materials (coated rotor).
4. A layered structure in the axial direction of the rotor may be made of appropriate
ferromagnetic and nonferromagnetic highconductivity materials (endring 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 highspeed applications when the perunit 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 perunit slip is 4 %, which means that 4 % of the
airgap power is lost in the rotor copper losses. If the rotating frequency is 200 Hz the
same absolute slip results in a 1 % perunit slip and, correspondingly, in a 1% per unit
rotor copper losses.
Solidrotor induction motors can be used as:
• Highspeed motors and generators.
• Two or threephase motors and generators for heavy duty, fluctuating loads,
reversible operating, and so forth.
• Highreliability motors and generators operating under conditions of high temperature,
high acceleration, active chemicals, and so on.
• Auxiliary motors for starting turboalternators.
• Flywheel applications.
26.
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.
• Eddycurrent 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 40’s till the 70’s. 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 solidrotor IM are not good enough.
Since the use of highspeed machines became more popular from the beginning of the 1990’s a
few FEM studies about solidrotor 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 solidrotor motors. The research has seven main objectives.
1) To create an analytical, multilayer transfermatrix 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 airgap 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 solidrotor 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 powerdependent 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 min1
and 12000 min1
. The main dimensions of the
250 kW – 300 kW prototype machines are: a 200 mm airgap diameter, a 280 mm stator stack
effective length.
27.
28
This work strongly focuses on the electromagnetic phenomena of the solidrotor machine,
irrespective of the fact that mechanic and thermodynamic studies are of essential importance,
especially as highspeed 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 wellknown multilayer transfermatrix method to be
used, especially, for the calculation of highspeed slitted solidrotor induction motors.
Improvement of the multilayer transfermatrix method was achieved by introducing
into the method a new endeffect factor and a new curvature factor for slitted solid
rotors equipped with wellconducting end rings. The new factors are functions of the
slit depths.
2. Definition of the best possible practical slitting of solid rotors equipped with well
conducting end rings for highspeed induction motors in the medium power range.
3. Definition of the best possible rotor active length to diameter ratio for slitted solidrotor
induction motors with wellconducting end rings.
4. Introducing of the powerdependent utilization factors for different types of solid
rotors.
5. Introduction of a new method to reduce the permeance harmonic content in the airgap
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:
28.
29
1. An analytical electromagnetic – and accurate enough  analysis of the solidrotor
induction machine is introduced. The method is very useful in everyday 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 endeffect 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 penetration
depth, and should thus be a function of the rotor slip frequency. In practice, in a deeply slitted
solid rotor with wellconducting 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 endring 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 highspeed 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 solidrotor induction motors have not been presented earlier. However, the
utilization factors for coppercoated solidrotor induction motors were presented by Gieras
(1995).
29.
30
1.6 Outline of the work
The multilayer transfermatrix 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 solidrotor 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 airgap 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.
30.
31
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 solidsteel rotors. Since the
conventional induction machine theory proved to be inadequate for solidrotor machines, the
need has grown to improve the methods of investigation. It has become necessary to determine
the solidrotor 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 its
magnetic and electric fields is a demanding process because the rotor material is magnetically
nonlinear and the electromagnetic fields are threedimensional. Thus, to solve the solidrotor
magnetic and electric fields fast and accurately enough is a demanding task. In the conventional
laminated squirrelcage 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 airgap 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 nondominant 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, threedimensional effects and nonlinearity 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 threedimensional FEM calculation, but it takes
far too much time to be used in everyday motor design proceeding. Besides, the modelling of a
rotation movement even more complicates the FEM calculation. Therefore, a threedimensional
31.
32
analytical solution for the rotor fields has to be found. The ultimate simplification is to solve the
Maxwell’s 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
1950’s, 1960’s and 1970’s. 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
rectangular BHcurve. 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 nonlaminated
steel rotors is far less than 45°.
An important feature of the solidrotor 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 nonlinear theory of the flux penetration into the solidrotor
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 BHcurve is
good only at very high levels of magnetisation. This analysis gives a constant value of 26.6° to
the 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 nonlinear 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 nonlinear 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 solidrotor
steel, in this particular case, are beneficial since they increase the solidrotor power factor. The
equivalent circuit approach was used by McConnell (1953), Wood (1960a), Angst (1962),
32.
33
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 nonlinear function for the BHcurve must be used.
The nonlinear variation of the fundamental B1H –curve is included in its entirety by
substituting the equation B1=cH(12/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 of H, 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 Pillai’s 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 transfermatrix 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 multilayer transfermatrix 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 nonisotropy or the end
effects, but the method gives consideration to the nonlinearity 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. (Pyrhönen
1991a).
33.
34
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 constant
permeability
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 Fourierseries. This method was first used by
Bondi (1957) and later developed by Yee (1971). The linear method requires solving of
Maxwell’s 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 airgap does not vary in the radial direction. The
mistake made here is negligible when the airgap 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 zaxis is taken in the
axial direction. The yaxis is normal to the rotor surface and the xaxis 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 negative xaxis, its position in the stator coordinates can be written as
p
r
tsxx s
RS
)1( ω−−= , (2.1)
34.
35
where p is number of pole pairs, r is rotor radius, t is time and ωs is stator angular speed.
y
x
z
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 the
machine
p
a
τ
π
= , where τp is pole pitch,
p
D
p
2
π
=τ . (2.2)
Equation (2.1) can be rewritten now
tsaxtax s
R
s
S
ωω +=+ (2.3)
Henceforth, the superscript R, which indicates to coordinate fixed to the rotor, will be left out.
The differential forms of Maxwell’s equations have to be used as a starting point. Ampere’s law
relates the magnetic field strength H with the electrical current density J and the electric flux
density D. Faraday’s induction law determines the connection between the electric field
strength E and the magnetic flux density B. Gauss’ equations definitely reveal that the
divergence of B is zero and the divergence of D is charge density ρ, i.e. B has no source and D
has the source and the drain.
t∂
∂ D
JH +=×∇ , ( 2.4)
35.
36
t∂
∂ B
E −=×∇ , (2.5)
0=⋅∇ B , (2.6)
ρ=⋅∇ D , (2.7)
The latter part of equation (2.4) representing Maxwell’s displacement current is omitted,
because the problem is assumed to be quasistatic, i.e. Maxwell’s displacement current is
negligible compared with the conducting current at frequencies which are studied in solidrotor
materials, see App. C.
In addition, the material equations are needed:
ED ε= , (2.8)
HB µ= , (2.9)
EJ σ= , (2.10)
where ε is the material permittivity, µ is the permeability of the material and σ its conductivity.
A twodimensional eddycurrent problem can be formulated in terms of the magnetic vector
potential A, 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 density B 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
36.
37
0=⋅∇ A . (2.12)
When equation (2.11) is substituted to Faraday’s law equation (2.5) we get
0=
∂
∂
+×∇ A
t
E . (2.13)
The sentence in parenthesis has no curl and may thus be written as a gradient of a scalar
function −φ. Now, the electric field strength can be written in the following form
φ
∂
∂
∇−−=
t
A
E . (2.14)
The charge density ρ can be assumed to be negligible in wellconducting solidrotor material.
Therefore, the divergence of the electric field strength is zero. The reduced scalar potential φ
describes the nonrotational part of the electric field strength. The nonrotational part is due to
electric charges and polarisation of dielectric materials. However, in a twodimensional 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 µ and conductivity σ
as constants, equation (2.4) can be written
t∂
∂
−=∇−⋅∇∇=×∇×∇
A
AAA µσ2
)()( . (2.15)
When the Cartesian coordinates are used and the Coulomb gauge, equation (2.12), is valid, the
differential equation of A can be expressed by
t
A
z
A
y
A
x
A iiii
∂
∂
µσ
∂
∂
∂
∂
∂
∂
=++ 2
2
2
2
2
2
, (2.16)
where i is x, y, or z (Yee 1971).
Because all fields in the induction machine may be assumed to vary sinusoidally as a function
of time, a steady state timeharmonic solution may be found in the analysis. The vector
potential A is considered. It can be expressed in a timeharmonic form by
37.
38
[ ]ts
zyxtzyx sj
e),,(Re),,,( ω
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 potential A to vary in the direction of the x
axis both as a function of place x with the term ejax
and as a function of time t with the term
ej ss tω
. The vector potential is obtained in form of a complex vector function
[ ])(j s
e),(Re),,,( tsax
zytzyx ω+
= AA . (2.18)
Now, equation (2.16) can be written as a complex exponent function
i
ii
Aa
z
A
y
A
)(
22
2
2
2
2
λ
∂
∂
∂
∂
+=+ , (2.19)
where
p
s
j2
j
d
s == µσωλ , (2.20)
dp is the penetration depth and λ 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 ωj+=×∇ . (2.21)
Using the annotation γ, which describes the variation of the fields in the axial direction, and δ
for the airgap length we get
r
2
δµ
λ
γ += a . (2.22)
Pyrhönen (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.
38.
39
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 in xyplane.
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 zdirection are:
(Pyrhönen 1991a)
)(j s
e
)
2
sinh(
)sinh(
)ee(
)
2
sinh(
)sinh(
e tsaxyayy
x
L
z
L
z
GA ωλλ
λ
λ
γ
γ +
−+= , (2.23)
)(j s
e
)
2
sinh(
)sinh(
)ee(j tsaxayy
y
L
z
GA ωλ
λ
λ +
−= , (2.24)
)(j s
e
)
2
sinh(
)cosh(
)
2
coth()
2
coth(ej tsaxy
z
L
zaLaL
GA ωλ
γ
γ
γ
γ
γ
λ +
−+= , (2.25)
where
++
−
=
)
2
coth()
2
coth()(
ˆj
r
2
00S
LaL
a
KI
G
γ
γ
λ
µ
λ
δ
µ
, Na
p
m
K ξ
π
0 = . (2.26)
In the rotor the magnetic flux density equations are:
( ) )(j s
e
)
2
sinh(
)cosh(
ee
)
2
sinh(
)cosh(
)
2
coth()
2
coth(ej tsaxyayy
x
L
z
L
zaLaL
GB ωλλ
λ
λ
γ
γ
γ
γ
γ
λλ +
−+
−+= ,
(2.27)
39.
40
( ) )(j s
e
)
2
sinh(
)cosh(
ee
)
2
sinh(
)cosh(
)
2
coth()
2
coth(e tsaxyayy
y
L
z
aL
za
a
LaL
GaB ωλλ
λ
λλ
γ
γ
γ
γ
γ
γ
λ +
−+
−++= ,
(2.28)
)(j s
e
)
2
sinh(
)sinh(
)
2
sinh(
)sinh(
e tsaxy
z
L
z
L
z
GB ωλ
λ
λ
γ
γ
λ +
−−= . (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:
)(j
0
s
e
)
2
sinh(
)cosh(
1
)
2
sinh(
)cosh(
)
2
coth()
2
coth(jd tsax
xx
L
z
aL
zaLaL
GyBΦ ω
λ
λλ
γ
γ
γ
γ
γ
λ +
∞−
−+−+== ∫ , (2.30)
)(j
0
s
e
)
2
sinh(
)sinh(
)
2
sinh(
)sinh(
d tsax
zz
L
z
L
z
GyBΦ ω
λ
λ
γ
γ +
∞−
−−== ∫ . (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 and Hz 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.
40.
41
AZ
Ax
HZ
ΦZ
Hx
Φx
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 the Az, Hx and Φx values at z = 0 (Yee 1971). a) Magnetic vector
potential at y = 0, b) magnetic field strength at y = 0, c) magnetic flux perunit length.
2.2 Calculation of a saturated solidrotor
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 multilayer
model and the coordinates used, Greig (1967).
In general, the current sheet
{ })(j s
e'Re tax
JJ ω+
= , (2.32)
lies between any two layers. Regions 1…K are layers made of material with resistivity ρk and
relative permeability µk. 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.
41.
42
K B
H
ρ µK K K1
K1
K1 B
H
ρ µK1 K1 K2
K2
B
Hk+1
k+1
B
H
ρ µ
k
k
k B
H
ρ µk k k1
k1 ρ µk1 k1
y
x
z
H J'k
3
B
H
ρ µ3 3
2
2
2 B
H
ρ µ2 2 1
1
1 ρ µ1 1
.
.
.
.
.
y = g K1
y = g 1
y = g 2
k1
k+1k+1
k+1
.
Fig. 2.3. Original twodimensional multilayer model (Greig 1967).
A stationary reference frame is chosen in which the exciting field travels with velocity ωs/a. A
region k, in which the slip angular speed is ωk = skωs, is therefore travelling at velocity (1
sk)ωs/a relative to the stationary reference frame (Greig 1967). Please note that in all the rotor
regions the slip sk is the same and a constant. In the stator regions the slip is zero.
Consider a general region k of thickness dk, as it is given in Fig. 2.4. The normal component of
the flux density on the lower boundary is By,k1, and the tangential component of the magnetic
field strength is Hx,k1. 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
42.
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 the x direction, and of finite thickness in the y direction.
Maxwell’s 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 at y = ±∞ .
3. If a current sheet exists between two regions, then '1 JHH kk −= − .
region k + 1 B
H
ρ µ
k+1 k+1 k
k
region k B
H
ρ µ
k k k1
k1
y = gk
region k  1
y = gk1
d k
ωk
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
B
T
ΥΥβ
Υ
β
Υ
, (2.33)
where
k
k
k
a µµ
Υ
β
0j
= and kkkk sa σµµωΥ 0s
2
j+= (2.34)
and [Tk], following Pipes (1956), is the transfer matrix for the region k. In the top region on the
boundary gK
1,1, −− −= KyKKx BH β . (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,,
1
e yg
KyKy
KK
BB −
−
−
= Υ
, (2.36)
43.
44
)(
1,,
1
e yg
KxKKx
KK
HH −
−
−
−= Υ
β . (2.37)
Solving the field in the bottom region on the boundary g1
1,11, yx BH β= . (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,
11
e gy
yy BB −
= Υ
, (2.39)
)(
1,1,
11
e gy
xx HH −
= Υ
. (2.40)
The transfer matrix can be used as follows, considering the boundary conditions (1) and (3).
The current sheet lies between regions k and (k+1). (Greig 1967).
[ ][ ] [ ]
=
−
1,
1,
21
,
,
x
y
kk
kx
ky
H
B
H
B
TTT L , (2.41)
[ ][ ] [ ]
−
=
+−−
−
−
',
,
121
1,
1,
JH
B
H
B
kx
ky
kKK
Kx
Ky
TTT 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 the
tangential field strength Hx,1 at the inner rotor boundary. The transfer matrix technique then
evaluates By,k and Hx,k at all interlayer 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 airgap and the stator. Iterative adjustment
of Hx,1 is made to adapt the conditions at the rotor surface.
As By,k and Hx,k are resolved at all interregion 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)
44.
45
The timeaverage 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 the zdirection and it may be written as:
ky
k
kz B
a
E ,,
ω
−= . (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
2
Re kxkykxkyk HBHB
a
P
ω
. (2.46)
The mechanical power density evolved by the region under slip sk is (Greig 1967)
)1(.mech kkk sPP −= . (2.47)
The ohmic loss I2
R elaborated by the region is (Greig 1967)
kkkk PsPP =− ,mech . (2.48)
2.2.1 Definition of the fundamental permeability in a nonlinear material
In a saturable material sinusoidally varying magnetic field strength creates a nonsinusoidal
magnetic flux density (Bergmann 1982). The amplitude spectrum of this flux density can be
numerically defined with the DCmagnetisation curve of the material. Fig. 2.5 shows how the
flattened B(ωt)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 1
ˆB of the Fourier series of the flux density is obtained by a numerical
integration:
45.
46
∫=
π
0
1 )(d)sin()(
π
2ˆ tttBB ωωω . (2.49)
The fundamental permeability of a particular working point is defined as
H
B
H
ˆ
ˆ
)ˆ( 1
1 =µ . (2.50)
B
H
H
ω t
H(ω t)
B (ω t)
B(ω t)
1
B1
ω t
B
H
Fig. 2.5. The definition of the fundamental magnetic flux density B1(ωt) produced by an external
impressed sinusoidally alternating magnetic field strength H(ωt) and the B1H curve with DC
magnetizing curve.
2.2.2 Rotor impedance
The rotor fundamental magnetomotive force in the airgap, referred to the stator, is
a
H
xHI
p
Nm xax
x
p
j
2
de'
2π
4
2 R
0
j
RR1 === ∫−τ
ξ
Θ , (2.51)
46.
47
from which the rotor current referred to the stator is found:
xH
Nam
p
I RR
2
jπ
'
ξ
−
= . (2.52)
The airgap flux of the machine is obtained by integrating the radial flux density at the rotor
surface over a pole pitch. Faraday’s induction law gives an equation for the rotor voltage per
phase referred to the stator:
y
ax
y B
a
LN
xLB
N
U
p
p
R
s
2
2

j
RsR
2
2
jde
2
j'
ξωξ
ω
τ
τ
−=−= ∫ . (2.53)
Finally, the rotor impedance referred to the stator is found:
x
y
H
B
p
mLN
I
U
Z
R
R
2
s
R
R
R
π
)(2
'
'
'
ξω
== . (2.54)
2.3 Effects of axial slits in a solid rotor
The performance of an induction machine with a solidsteel 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 nonisotropy 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 threedimensional 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 solidrotor induction motor
by means of a FEM program, but only a few rotor slit parameters are used. Also Laporte (1994)
47.
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 τu, a slit and a tooth width wu and wt, relative
permeability of the tooth µt and both slit and tooth resistivity ρu ja ρt, Fig. 2.6 (Freeman 1968).
Here, it is assumed that the slit is not of a magnetic medium, i.e. µu = 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
t
t
ττ
µµ
ww
y += , (2.55)
tut
ut
µ
τµ
µ
ww
x
+
= , (2.56)
tuut
utu
ww ρρ
τρρ
ρ
+
= . (2.57)
wt wu
τu
y
x
z
Fig. 2.6. Slitted solidrotor surface.
48.
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 threedimensional
field problem under constant permeability. Yee (1971), too, solves the threedimensional field
problem under constant permeability. This kind of approach is usually limited because of the
saturation in the stator teeth and rotor end areas (Yee 1972). Ducreux (1995) calculated the end
effects of a solid rotor by means of the 2D and 3D FEM program. He also compared the 3D
results with the 2D results, which were corrected by using correction factors given by Yee
(1971) and Russell (1958).
2.4.1 Solid rotor equipped with highconductivity end rings
If the solid rotor is equipped with end rings made of a highconductivity material, e.g. copper or
aluminium, the rotor end effects, in many of the studies, are considered to be diminutive and
they have been ignored; but, according to this study, the end effects should also be considered
when wellconducting end rings are used. For a solid rotor with end rings it is possible to obtain
fairly accurate calculations by using an equivalent conductivity for the rotor material. The
49.
50
equivalent conductivity takes the resistivity of the end rings into account when the rotor
conductivity is considered. This technique was studied by Russell (1958), Jamieson (1968a),
Rajagopalan (1969), Yee (1971), Woolley (1973), and Jinning (1987). The leakage inductance
of the end rings can be ignored as infinitesimal. In other words, the rotor is analysed as being
infinitely long, and the resistivity of the end rings is added to the resistivity of the rotor core
steel. The analyses obtained by this method are very congruent to the measured results.
Russell (1958) suggested that the actual loss in the rotor surface shell could be evaluated by
assuming all the currents to be axial, but that the resistivity of the shell is increased by a factor
)
2
π
tanh(
π
2
1
1
p
p L
L τ
τ
α
−
= . (2.58)
Further based on this, a general endeffect factor applicable for both the solid and slitted rotors
can be chosen as,
)1(1er −+= αCK , (2.59)
where C = 1 for rotors without end rings,
C = 0.3 for thick copper end rings.
Woolley (1973) defined the endeffect correction factor in the following way,
2
R
R
1
2
11er )tanh(4
2
1
++=
D
pL
kQQK , (2.60)
where )tanh()(1
R
R
1
R
R
1
D
pL
k
pL
D
Q +−= . (2.61)
where
erc
cer
1
ρ
ρ
t
t
k = , and ter and ρer represent the end region effective thickness and the resistivity
and tc and ρc represent the cylindrical shell region effective thickness and resistivity,
respectively. If the rotor is slitted, the slit depth can be used for tc, otherwise an appropriate
value for tc seems to be the depth of the flux penetration δp in the surface of the rotor. If the end
50.
51
rings are made of nonmagnetic material with a thickness greater than the characteristic
penetration depth dp in that material, the value of dp should be used for ter. Otherwise, the end
ring thickness should be used (Woolley 1973).
If the dimensions of the low resistivity end rings are known, the endeffect factor can also be
defined as follows; the teeth in the rotor steel act as rotor bars, where the rotor fundamental
current flows, assuming deep enough slits. The endeffect factor for the rotor resistivity is
derived as a ratio between a rotor tooth resistance and a total rotor phase resistance (Huppunen
2000b).
By using the tooth length LR, the conductivity of the tooth σr and tooth crosssection area At the
DC resistance of the rotor tooth may be written as
tR
R
tR
A
L
R
σ
= . (2.62)
The resistance of the end ring in a tooth pitch is by the average diameter of the end ring Der, the
conductivity of the end ring σer, the cross area of the end ring Aer and the number of the rotor
teeth QR
Rerer
er
er
π
QA
D
R
σ
= . (2.63)
When a tooth current is marked as IsR, the endring current is (Richter, 1954)
=
R
sR
er
π
sin2
Q
p
I
I . (2.64)
The currents cause copper losses in a rotor
)2( 2
erer
2
sRtRRRCu, IRIRQP += . (2.65)
In a twopole rotor the number of phases is equal to the number of teeth, thus the resistance of
the rotor phase is
51.
52
+=
R
2
er
tRR
π
sin2
Q
p
R
RR . (2.66)
The endeffect factor is defined as a ratio between the resistance of the rotor tooth RtR and the
resistance of the rotor phase RR:
R
sR
er
R
R
K = . (2.67)
The described method sets the values for the endeffect factor between [0.5 … 0.7] when a
copper squirrel cage is used and between [0.7 … 0.9] for a solidsteel rotor with copper end
rings. These values indicate that even when a solidsteel rotor with end ring is considered, the
end effects must be taken into account.
2.4.2 Solid rotor without end rings
When the solid rotor is not equipped with wellconducting end rings, the rotor end fields have a
significant effect on the motor characteristics. It would also be possible to use a correction
factor for the rotor impedance as this rotor structure is considered. Wood (1960c), Angst
(1962), Yee (1971), Woolley (1973) proposed complex correction factors applicable to the
effective rotor impedance.
Yee (1971) proposes a finite length factor for the effect of finite rotor length:
2
er
2
2
coth
2
coth
2
1)(
γ
γ
γ
λ
L
aLaL
aLsK
−
+
+= . (2.68)
This factor takes also the loading into account. Ker(s) is analogous to the endeffect factor
derived by Angst (1962). Furthermore, Yee (1971) declares that arg (Ker) is found to be very
small, thus, for typical solidrotor machines, Ker can be simplified to a real constant. Except for
very small slip values, coth (λL/2) ≈ 1. Setting, in addition, γ = a,
52.
53
2
2
coth1
2
coth1
er
−
+
+
=
La
aL
La
aL
K . (2.69)
Another theory proposed for the calculating of the end effects in a finitelength solid rotor
without end rings assumes that the rotor flux can be divided into two components, Fig. 2.7. Flux
Φ1 enters the rotor at the airgap and follows a circumferential path near the airgap. Flux Φ2
enters the rotor at the airgap and follows an axial path near the airgap and then a path across
the end faces. Flux Φ1 is associated with the most heavily saturated parts of the rotor, while flux
Φ2 follows relatively unsaturated parts in the rotor, when the machine is rotating at its normal
working range of slip. Flux Φ1 corresponds to the main axial eddy currents, and flux Φ2 to the
end currents. In a rotor fitted with low resistance end rings, flux Φ2 is greatly reduced in the
magnitude (Yee 1972).
The aim of the following analysis is to derive the rotor impedance for a partly saturated rotor by
using the MLTM method to describe the electromagnetic fields associated with flux Φ1, and by
using the linear theory to describe the fields associated with flux Φ2. An analysis combining
these two methods was introduced by Pyrhönen (1991a). In the following the solution for the
end fields is given. The equations are given earlier by Yee (1972).
Φ Φ1 2
a) b)
Fig. 2.7. Components of the flux in a twopole rotor. a) Φ1 corresponds to the axial eddy currents and b)
Φ2 to the end currents.
53.
54
The equations (2.23)  (2.29) give the rotor fields in rotor coordinates when a constant magnetic
permeability is assumed. The rotor fields Ex(y=0), Ey(z=±L/2), Ex(z=±L/2), Hz(y=0), Hy(z=±L/2), Hx(z=±L/2)
associated with Φ2 are defined directly from these equations since the flux Φ2 follows the
unsaturated parts of the rotor. Ez(y=0) and Φ1 are defined from Hx(y=0) assuming that the magnetic
properties of the material can be described using the multilayer transfermatrix method.
Using equation (2.1) the equations may be expressed with respect to the stator coordinates. The
xcoordinate in stator reference frame is marked as x1.An annotation H0 is used.
λ
µ
GH
1
0 = . (2.70)
In addition, the following algebraic approximations are made as the loss of accuracy is
negligible: a>>λ and γλ >> .
Using equations (2.27) – (2.29) for the flux densities also gives the magnetic field strengths.
Notifying that the phase angle of the imaginary unit is π/2 and the phase angle of the λ is π/4, it
can be written (Pyrhönen 1991a):
−+= +
=
)
2
sinh(
)cosh(
)
2
coth()
2
coth(ee )(j4
π3
j
00
s1
L
zaLaL
HH tax
yx
γ
γ
γ
γ
γ
λω
, (2.71)
−−= +
=
)
2
sinh(
)sinh(
)
2
sinh(
)sinh(
ee )(j4
π
j
00
s1
L
z
L
z
HH tax
yz
λ
λ
γ
γω
, (2.72)
)
2
coth(eee )(j4
π3
j
0
2
s1
L
HH aytax
L
zx λω+
=
= , (2.73)
aytax
L
zy HH eee )(j4
π
j
0
2
s1 ω+
=
= . (2.74)
54.
55
The respective electric field strengths just outside the rotor surface are found by deriving
equations (2.24) – (2.26) and by substituting the values of y and z: (Pyrhönen 1991a)
)
2
sinh(
)sinh(
e/e )(j
lin0s
2
π
j
00
s1
L
z
sHE tax
yx
γ
γ
ρµµω ω+
=
−= , (2.75)
)(j
lin0s
2
π
j
0
2
s1
ee/e taxay
L
zx sHE ω
ρµµω +
=
−= , (2.76)
)(j
lin0s0
2
s1
ee/ taxay
L
zy sHE ω
ρµµω +
=
−= . (2.77)
The saturated components Hx(y=0) and Ez(y=0) are defined by the nonlinear MLTM method,
equation (2.42), when the electric field strength in zdirection at the surface of the rotor
according to equation (2.45) is
R
s
0 yyz B
a
s
E
ω
==
. (2.78)
Φ2 can be obtained by integrating, over the surface y=0, that component of By(y=0) which
corresponds to the tangential electric field strength Ex(y=0). The curl equation of the electric field
strength gives
t
B
z
E
x
E yxz
∂
∂
=
∂
∂
−
∂
∂
. (2.79)
By choosing only the component that corresponds to the flux Φ2 equations (2.75) and (2.79)
give
∫
+
=
∂
∂
=
)
2
sinh(
)cosh(
e/ed )(j
0
2
π
j
0 1
2 L
z
s
j
H
t
z
E
B tax
lins
s
Φ
s
γ
γ
γρµµω
ω
ω
. (2.80)
By integrating over the surface y = 0, the unsaturated path flux Φ2 is obtained as
55.
56
sH
a
xzBΦ
p
p
L
L
Φ /
4
dd lin0s0
s
2
2
2
2
2 2
ρµµω
ω
τ
τ
== ∫ ∫
− −
. (2.81)
The airgap voltage of the machine is calculated with Faraday’s induction law
)(
2
j 21s ΦΦ
N
U +−=
ξ
ωδ . (2.82)
By using the complex Poynting vector, see App. E, the average power density flow into the
surface can be defined as
*
2
1
HES ×= . (2.83)
The complex power that flows into the rotor is found by integrating the Poynting vector over all
the rotor surfaces (Yee 1972). By using equations (2.71) – (2.77), we obtain
.d)(
2
1
d)(
2
1
d)(
2
1
d)(
2
1
π'
0

0

2/
*
2/2/
*
2/
2/
2/
2/
2/
0
*
00
*
0
−+
+−=
∫ ∫
∫ ∫
∞ ∞
====
− −
====
yHEyHE
zHEzHED
Lz
yLzx
Lz
xLzy
L
L
L
L
y
zyx
y
xyzS
(2.84)
In equation (2.84) the field variations in the direction of the xcoordinate have already been
integrated and the result is included in the term πD. However, the terms in (2.84) have
maintained their original form for convenience. This method gives fairly accurate results when
the machine is running at low slips, since then flux Φ2 is unsaturated. From the present
theoretical model, it is evident that, as the stator current increases, the magnitude of flux Φ1 is
reduced compared to the magnitude of flux Φ2, since Φ1 is associated with the saturated region
of the rotor. Since Φ2 is concentrated near the ends of the rotor, the overall effect is a more
pronounced increase of the flux near the ends of the rotor (Yee 1972).
56.
57
2.5 Effect of the rotor curvature
The previously defined endeffect factor brings the calculation results closer to the measured
values, but the calculation gives still too much output power from the machine at a given slip.
Especially in slitted rotors, the curvature should be taken into consideration, since the rotor
teeth get narrower when proceeding towards the shaft. Wood (1960b) replaced hyperbolic
functions of the rectilinear model by complex Bessel function combinations and he used the
Kelvin functions to calculate the value of the complex Bessel functions. The effects of the
curvature were later studied by Freeman (1974), who analysed the solid rotor with the MLTM
method in polar coordinates. Kesavamurthy (1959) and Rajagopalan (1969) used a correction
factor, which increased the resistivity of the rotor. In the following, a correction factor for the
curvature is defined for slitted solid rotors when the MLTM method is used in the Cartesian
coordinates.
For the slotted solid rotors the substitute parameters for the permeability and the conductivity of
the rotor material were defined earlier, see equations (2.55)  (2.57). There, the rotor was
assumed to be rectangular, when the substitute parameters are constants in the rotor. In fact, the
tooth pitch and the cross area of the teeth decrease towards the negative ydirection, i.e. from
rotor surface towards the shaft. At the same time, the substitute material parameters, i.e. the
permeability and the conductivity, alter, Fig. 2.8. The darkened area in the figure describes the
cross section of the rotor tooth in a calculation layer of the MLTM method. The curvature of the
rotor can be taken into account by calculating the curvature factors for the substitute parameters
in each calculation layer. The curvature factors have to be defined separately for both the tooth
pitch and the tooth width, since they vary in a different relation. Using the diameter of the rotor
DR and the distance from the axis to the calculation region boundary gk, the curvature factor for
the slit pitch KC,k may be obtained as (Huppunen 2000b)
R
R
C,
2
1
D
gD
K k
k
−
−= . (2.85)
The slit pitch in the calculation region k is uC,u,' ττ ⋅= kk K , (2.86)
and the tooth width is uuC,t,' wKw kk −⋅= τ . (2.87)
Now, the equations (2.55)  (2.57) may be rewritten, as the curvature is taken into consideration.
57.
58
kk
k ww
k
u,
u
u,
t,
ty
''
'
)(
ττ
µµ += , (2.88)
tut,
u,t
x
'
'
)(
µ
τµ
µ
ww
k
k
k
+
= , (2.89)
k
k
ww
k
t,uut
u,tu
'
'
)(
ρρ
τρρ
ρ
+
= . (2.90)
g
Fig. 2.8. Effects of the curvature to the crosssection area of the tooth in slitted solid rotors.
The field calculation can also be executed in the polar coordinates by the multilayer transfer
matrix method, and thus the curvature effects are taken into account. The multilayer model is
illustrated in Fig. 2.9.
K
K1
k+1
k
k1
2
1
HK1HK Hk+1 Hk Hk1 H2 H1
r
r
r
r
r
rK1
k+1
k
k1
2
1
E
E
E
E
E
E
1
2
k1
k
k+1
K1
BK1
B2
B1
B
B
B
k1
k
k+1
r
z
Fig. 2.9. The crosssection through a Klayer cylindrical induction device.
58.
59
The model is assumed to be infinitely long in the zdirection, so the end effects have to be taken
into consideration by the endeffect factor or the linear endfield calculation. Now, the transfer
matrix between each region k is according to Freeman (1974)
[ ]
=
=
−
−
−
−
1,
1,
1,
1,
,
,
kx
ky
k
kx
ky
kk
kk
kx
ky
H
B
H
B
db
ca
H
B
T , (2.91)
where { })(')()(')( 12121 βββββ νννν IKKIak −−= , (2.92)
{ })()()()( 12121 ββββσ νννν KIIKrb kkk −−= − , (2.93)
{ })()()()(j 1212,1 ββββωµ ννννφ KIIKrc kkk −−= − , (2.94)
{ })()(')()(' 12121 βββββ νννν KIIKdk −−= , (2.95)
kp
k
d
r
,
1
1
j−
=β , (2.96)
kp
k
d
r
,
2
j
=β , (2.97)
kr
k
p
,
,
µ
µ
ν φ
= , (2.98)
and [Tk] is called transfer matrix for region k. µφ,k and µr,k are the permeability of the layer k in
the φ and r directions respectively. The Bessel functions are of the modified first and second
kind, of the order ν.
2.6 Computation procedure developed during the work
The practical analysis in this work is based on the MTLM method. The MLTM analysis was
programmed to compute the electromagnetic field quantities and power flow at any boundaries
between all layers, once By,k or Hx,k is given at any particular boundary. The procedure uses the
rectangular multilayer model of the rotor (Fig. 2.3) and it is commenced by assuming a low
59.
60
value of tangential field strength Hx,1 at the inner rotor surface and by calculating the
corresponding normal component of flux density B1. The MLTM technique then evaluated By,k
and Hx,k at all rotor interlayer boundaries up to the rotor outer surface. At this interface, where
Hx,k corresponds to the total rotor current, the model was connected to a conventional equivalent
circuit representation of the airgap and the stator. Iterative adjustment of H1 was then used to
attain a specified machine operation condition.
To take into account the nonlinear magnetization characteristic of a solidsteel medium such a
medium was divided into a number of thin layers. The permeability of each layer was
considered to be corresponding to the tangential magnetizing field in the preceding layer. The
BHcurve of the steel was represented by 30 data points and an interpolation routine was used to
find B and, hence, the permeability at any given value of H. In a typical case, a 100 mm thick
steel rotor was divided into 500 layers.
The slitted rotor section was modelled by a nonisotropic region with substitute parameters per
slit pitch for the permeability and the conductivity of the steel medium. This scheme leads to a
solution, where the field distribution is equal in slits and teeth regions. However, this is an
assumption that does not meet the real facts and, must therefore be considered carefully. If the
slit geometry becomes more complicated than a rectangular shape or the ratio of slit and tooth
widths become very low or large, the assumption may break down.
It is often assumed that the effect of the rotor curvature may be neglected in the analysis of a
solid rotor. This is, however, a supposition that is valid only for smooth solid rotors, where the
penetration depth is much lower than the rotor radius. But, in slitted rotors consideration must
be given to the curvature because the slits force the flux to penetrate deeper than the slit depth
is. Here, the curvature effect was catered by calculating the substitute parameters of slitting in
each layer again.
The field phenomena in a solid rotor form a threedimensional problem which must be taken
into account in the analysis. When wellconducting end rings are used (copper or aluminum
alloys) the current paths in the slitted rotor region are nearly axial and the tangential current
flow occurs mainly in the endring regions. In such a case, the end effects of the rotor can be
taken into account by decreasing the conductivity of the rotor medium in such a way that the
total conductivity in a current path has been lumped into the stator active length. In this thesis, it
is focused on copperendring solid rotors, hence the method described above has been used.
60.
61
But also solid rotor induction motors without separate end rings have been designed and tested.
For that reason, the study treats the theory which considers the rotor magnetizing flux by
dividing it into two components Φ1 and Φ2 and which is originally introduced by Yee (1972).
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