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Lappeenrannan teknillinen yliopisto Lappeenranta University of Technology
Jussi Huppunen
HIGH-SPEED SOLID-ROTOR INDUCTION MACHINE – ELECTROMAGNETIC CALCULATION AND DESIGN
Thesis for the degree of Doctor of Science (Technology) to be presented with due permission for public examination and criticism in the Auditorium 1382 at Lappeenranta University of Technology, Lappeenranta, Finland on the 3rd of December, 2004, at noon.
Acta Universitatis Lappeenrantaensis 197
ISBN 951-764-981-9 ISBN 951-764-944-4 (PDF)
ISSN 1456-4491
Lappeenrannan teknillinen yliopisto Digipaino 2004
ABSTRACT Jussi Huppunen High-Speed Solid-Rotor Induction Machine – Electromagnetic Calculation and Design Lappeenranta 2004 168 p. Acta Universitatis Lappeenrantaensis 197 Diss. Lappeenranta University of Technology ISBN 951-764-981-9, ISBN 951-764-944-4 (PDF), ISSN 1456-4491. Within the latest decade high-speed motor technology has been increasingly commonly applied
within the range of medium and large power. More particularly, applications like such involved
with gas movement and compression seem to be the most important area in which high-speed
machines are used.
In manufacturing the induction motor rotor core of one single piece of steel it is possible to
achieve an extremely rigid rotor construction for the high-speed motor. In a mechanical sense,
the solid rotor may be the best possible rotor construction. Unfortunately, the electromagnetic
properties of a solid rotor are poorer than the properties of the traditional laminated rotor of an
induction motor.
This thesis analyses methods for improving the electromagnetic properties of a solid-rotor
induction machine. The slip of the solid rotor is reduced notably if the solid rotor is 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 multi-layer transfer-matrix method is
developed especially for the calculation of axially slitted solid rotors equipped with well-
conducting end rings. The calculation results are verified by using the finite element analysis
and laboratory measurements. The prototype motors of 250 – 300 kW and 140 Hz were tested
to verify the results. Utilization factor data are given for several other prototypes the largest of
which delivers 1000 kW at 12000 min-1.
Keywords: high-speed induction machine, solid rotor, multi-layer transfer-matrix, harmonic losses. UDC 621.313.333 : 621.3.043.3
Acknowledgements
In 1996, at the Laboratory of Electrical Engineering, Lappeenranta University of Technology,
the research activities related to this thesis got started, being part of the project “Development
of High-Speed Motors and Drives”. The project was financed by the Laboratory of Electrical
Engineering, TEKES and Rotatek Finland Oy.
I wish to thank all the people involved in the process of this thesis. Especially, I wish to express
my gratitude to Professor Juha 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 pre-examiners 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
Contents
ABBREVIATIONS AND SYMBOLS.........................................................................................9 1. INTRODUCTION ...............................................................................................................15
1.1 APPLICATIONS OF HIGH-SPEED MACHINES.....................................................................18 1.2 HIGH-SPEED MACHINES..................................................................................................20 1.3 SOLID-ROTOR CONSTRUCTIONS IN HIGH-SPEED INDUCTION MACHINES ........................22 1.4 OBJECTIVES OF THE WORK .............................................................................................27 1.5 SCIENTIFIC CONTRIBUTION OF THE WORK......................................................................28 1.6 OUTLINE OF THE WORK ..................................................................................................30
2. SOLUTION OF THE ELECTROMAGNETIC FIELDS IN A SOLID ROTOR .......31 2.1 SOLUTION OF THE ELECTROMAGNETIC ROTOR FIELDS UNDER CONSTANT PERMEABILITY
34 2.2 CALCULATION OF A SATURATED SOLID-ROTOR.............................................................41
2.2.1 Definition of the fundamental permeability in a non-linear 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 high-conductivity end rings........................................49 2.4.2 Solid rotor without end rings.................................................................................52
2.5 EFFECT OF THE ROTOR CURVATURE...............................................................................57 2.6 COMPUTATION PROCEDURE DEVELOPED DURING THE WORK........................................59
3. ON THE LOSSES IN SOLID-ROTOR MACHINES.....................................................62 3.1 HARMONIC LOSSES ON THE ROTOR SURFACE.................................................................63
3.1.1 Winding harmonics ...............................................................................................63 3.1.2 Permeance harmonics............................................................................................69 3.1.3 Decreasing the effect of the air-gap harmonics ....................................................76 3.1.4 Frequency converter induced rotor surface losses................................................86
3.2 FRICTION LOSSES............................................................................................................87 3.3 STATOR CORE LOSSES ....................................................................................................90
3.3.1 Stator lamination in high-speed machines ............................................................94 3.4 RESISTIVE LOSSES OF THE STATOR WINDING .................................................................94 3.5 LOSS DISTRIBUTION AND OPTIMAL FLUX DENSITY IN A SOLID-ROTOR HIGH-SPEED
MACHINE ........................................................................................................................96 3.6 RECAPITULATION OF THIS CHAPTER ..............................................................................97
4. ELECTROMAGNETIC DESIGN OF A SOLID-ROTOR INDUCTION MOTOR ..99 4.1 MAIN DIMENSIONS OF A SOLID-ROTOR INDUCTION MOTOR ...........................................99
4.1.1 Utilization factor....................................................................................................99 4.1.2 Selection of the L/D-ratio....................................................................................103 4.1.3 Slitted rotor with copper end rings......................................................................104 4.1.4 Effects of the end-ring dimensions .....................................................................108
4.2 DESIGN OF SLIT DIMENSIONS OF A SOLID ROTOR .........................................................109 4.2.1 Solving the magnetic fields of a solid-rotor induction motor by means of the
FEM-analysis.......................................................................................................110 4.2.2 FEM calculation results.......................................................................................115 4.2.3 Study of the rotor slitting ....................................................................................119 4.2.4 Comparison of the FEM with the MLTM method .............................................127
4.3 MEASURED RESULTS ....................................................................................................135 4.4 DISCUSSION OF THE RESULTS.......................................................................................136
5. CONCLUSION ..................................................................................................................138 5.1 DISCUSSION..................................................................................................................138 5.2 FUTURE WORK..............................................................................................................139 5.3 CONCLUSIONS ..............................................................................................................140
REFERENCES: .........................................................................................................................143 APPENDIX A.............................................................................................................................153 APPENDIX B .............................................................................................................................155 APPENDIX C.............................................................................................................................162 APPENDIX D.............................................................................................................................164 APPENDIX E .............................................................................................................................166
9
Abbreviations and symbols
Roman letters a abbreviation, function, number of parallel conductors, constant a1k factor for calculating the slot harmonic amplitudes A area, linear current density, vector potential Aj cross-section area of one conductor A magnetic vector potential (vector) b flux density, function, distance B magnetic flux density Bn magnitude of magnetic flux density drop c function, constant C constant, utilization factor CT torque coefficient d function dk thickness of layer dp penetration depth dc diameter of conductor D diameter, electric flux density E electric field strength, electromotive force (emf) Eew distance of the coil turn-end f frequency F function g boundary of 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
10
KC curvature factor Ker end-effect factor l length lm length of one turn of the winding L length L’ electrical length m number of phases n constant, number of coil turns in one slot N number of turns in series per stator phase o width of slot opening n unit normal vector p pole pairs, power P active power q number of slots per phase and pole qm mass flow rate Q function QR number of rotor 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
11
x function x, y, z coordinates X reactance Yk complex function of layer k Z impedance Greek letters α factor, end-effect factor, angle β complex function βδ flux distortion factor γ factor γ complex function, a measure of field variation in the axial direction δ air-gap 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 τ p pole pitch τ u slot pitch Φ magnetic flux χ chord factor ωs stator angular frequency Ω mechanical rotating angular speed
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 air-gap 0 basic value, initial value 1 fundamental, bottom layer
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 (time-harmonic presentation) a 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 multi-layer transfer-matrix 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 “high-speed”. In some occasions, the high speed is
determined by the machine peripheral speed. This can be justified from the mechanical
engineering point of view. Speeds over 150 m/s are considered to be high speeds (Jokinen
1988). This kind of a peripheral speed may, however, be reached with a two-pole, 50 Hz
machine which has a rotor diameter of 0.96 m. An electrical engineer may not regard a 50 Hz
machine as a high-speed machine. From the motor manufacturer’s point of view a two-pole
machine the supply frequency of which is considerably higher than the usual 50 Hz or 60 Hz is
normally considered to be a high-speed machine. However, some motor manufacturers have
called large 3600 min-1 machines high-speed machines. The difference of terms used in the
subject can be explained from the other viewpoint, which is that of the power electronics.
Present-day frequency converters are well able to produce frequencies up to a few hundreds of
hertz. However, the voltage quality of many converters is no more satisfactory if a purely
sinusoidal motor current is required. With respect to the present-day high-power IGBT-
technology the switching frequency is limited typically to 1.5 … 6 kHz. 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 present-day
industrial frequency converters are considered, frequencies in the range of 100 … 400 Hz
appear to be high frequencies. There are several research projects aiming at the design of ultra
high-speed machinery. For example, Aglen (2003) reported the application of an 80000 min-1
rotating permanent magnet generator to a micro-turbine and Spooner (2004) described the
project the objective of which was the design of a 6 kW, 120000 min-1 axial flux induction
machine to be applied to a turbo charger. This thesis, however, focuses on electric machines
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 high-speed machines, which are rotating at higher speeds than it would be
possible to directly reach by means of the network frequency, is to replace a mechanical
gearbox by an electrical one and attach a load-machinery directly on the motor shaft. This gives
also full speed control for the drive. The use of converters has become possible in the latest
decades as high switching frequency voltage source converters – often known as inverters –
have came into the market. Converters, however, cause extra heating problems even in normal
speed machines and thus a careful design combining the inverter with a solid-rotor machine is
needed.
The technology research in the field of high-speed machines has been particularly active in
Finland. Pyrhönen (1991a) studied ferromagnetic core materials in smooth solid rotors.
Lähteenmäki (2002) researched rotor designs and voltage sources suitable for high-speed
machines. His study focused on the design of squirrel cage and coated solid rotors. Saari (1998)
studied thermal analysis of high-speed induction machines and Kuosa (2003) analysed the air-
gap friction in high-speed machines. Antila (1998) and Lantto (1999) studied active magnetic
bearings used in high-speed induction machines. However, all of the above-mentioned studies
concentrated on machines running faster than 400 Hz. This thesis focuses on machines that run
at supply frequencies from 100 Hz to 300 Hz.
Also some other dissertations treating the solid rotor have been done. Peesel (1958) studied
experimentally slitted solid rotors in a 19 kW, 50 Hz, 4-pole induction motor. He manufactured
and tested 25 different rotors. Dorairaj (1967a; b; c) made experimental investigations on the
effects of axial slits, end rings and cage winding in a solid ferromagnetic rotor of a 3 hp, 50 Hz,
6-pole induction motor. Balarama Murty (Rajagopalan 1969) also studied the effects of axial
slits on the performance of induction machines with solid steel rotors. Wilson (1969) introduced
a theoretical approach to find out which is the impact of the permeability of the rotor material
on a 5 hp, 3200 Hz solid-rotor induction motor. Shalaby (1971) compared harmonic torques
produced by a 3.6 kW, 50 Hz, 4-pole induction machine with a laminated squirrel-cage rotor
and by the same machine with a solid rotor. Woolley (Woolley 1973) examined some new
designs of unlaminated rotors for induction machines. Zaim (Zaim 1999) studied also solid-
rotor concepts for induction machines.
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
high-speed solid-rotor induction motors. During the latest years research has been focused on
the improving of the efficiency of the high-speed solid-rotor motor construction. It has turned
out that, when a solid rotor is used, it is extremely important to take care of the flux density
distribution on the rotor surface. A perfectly sinusoidal rotor surface flux density distribution
produces the lowest possible losses. This is valid for both time dependent and spatial
harmonics. Because even a smooth solid construction high-speed steel rotor runs at quite a low
per-unit slip, this indicates that it is possible to reach a good efficiency if the stator losses and
the harmonic content on the air-gap flux and the rotor losses are kept low. Research has given
good results and the efficiencies of the high-speed motors have increased up to the level of the
efficiencies of typical 3000 min-1 commercial induction motors of the same output power.
At LUT, research in the field got started with the study on a 12 kW, 400 Hz induction machine
(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
squirrel-cage solid rotor were tested for milling machine applications (Pyrhönen 1996). Later, 8
kW, 300 Hz and 12 kW, 225 Hz copper squirrel-cage solid-rotor induction motors were
manufactured to be used in milling spindle machines.
The next stage brought the investigation of bigger machines. A 200 kW, 140 Hz slitted solid-
rotor induction machine and a 250 kW, 140 Hz slitted solid-rotor induction machine with
copper end rings were analyzed (Huppunen 1998a). Afterwards, several induction machines
with both rotor types in the power range of 150 kW – 1000 kW and in the supply frequency
range of 100 – 200 Hz were designed, manufactured and tested in co-operation with Rotatek
Finland Oy and LUT.
LUT has also cooperated in the developing of some permanent magnet high-speed machines.
Permanent magnet machines with output powers and rotational speeds of 20 kW, 24000 min-1
and 400 kW, 12000 min-1 (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.
18
Contrarily to this, the simple, rugged solid-rotor high-speed induction machine seems to be an
attractive solution for several industrial applications even though its efficiency is somewhat
lower and the size somewhat larger than the corresponding values of a PMSM at the same
performance.
Generally, the output torque of an electric machine is proportional to the product of the ampere-
turns and the magnetic flux per pole. Since the ampere-turns and the magnetic flux per pole
have limited values for a given motor size, the most effective way to increase the output power
is to drive the machine at a higher speed than normally.
The main advantages of using the motor in a high-speed range are the reduction of the motor
size and the absence of a mechanical gearbox and mechanical couplers. When using appropriate
materials the volume per power ratio and the weight per power ratio are nearly inversely
proportional to the rotating speed in the high-speed range. Thus, when the motor speed is near
10000 min-1, the motor size and the weight will decrease – depending on the cooling
arrangements – to about one third of the size of a conventional network frequency motor for
3000 min-1. This is valid for open motor constructions. If a totally closed construction is used
the benefit of the reduced motor size is lost.
Solid-rotor constructions are used because of mechanical reasons. This rotor type is the
strongest possible one and may be used in conjunction even with mechanical bearings at
elevated speeds since the rotor maintains its balance extremely well. When the load is directly
attached onto the solid-rotor shaft and elevated speed is used, the solid-rotor construction is still
able to achieve a sufficient mechanical strength and avoid balance fluctuations and vibrations,
which might damage the bearing system.
1.1 Applications of high-speed machines
High-speed solid-rotor induction motors may be used in power applications ranging from a few
kilowatts up to tens of megawatts. The main application area lies in the speed range where
laminated rotor constructions are not rigid enough as the mechanical viewpoint is considered.
Jokinen (1988) defined the speed limits for certain rotor types. The curves in Fig. 1.1 are
obtained, when conventional electric and magnetic loadings are used, the rotors are
manufactured of steel with a 700 MPa yield stress and the maximum operating speed is set 20
percent below the first critical speed. The rotational speed limit for the laminated rotors varies
19
from ca. 50 000 min-1 to 10 000 min-1 while the power increases from a few kilowatts to the
megawatt range. However, this speed level may demand several special constructions e.g. rotors
with no shaft and with FeCo-lamination as well as with CuCrZr-alloy bars. Also the upper
speed limit for the solid-rotor technology is set by the mechanical restrictions and is 100 000
min-1 to 20 000 min-1, respectively. But, these mechanical restrictions define the maximal speed
for a certain rotor volume. The limiting power, however, is always defined by the thermal
design of the machine.
10
100
1000
10000
1000 10000 100000
Rotational speed [rpm]
Max
imum
pow
er [k
W]
Laminated rotor
Solid rotor
Fig. 1.1. Powers limited by the rotor material yield stress (700 MPa) versus rotational speed (Jokinen
1988).
High-speed machines are mainly applied to blowers, fans, compressors, pumps, turbines and
spindle machines. The best efficiencies for these devices are achieved at elevated speeds, and
by using high-speed machines gearboxes and couplings can be avoided. The biggest potential
for high-speed machines lies on the field of turbo-machinery. Potential applications are blowers,
fans, gas compressors and gas turbines, because the rotational speeds of the gas compression
units are typically high. A common way to manufacture a gas compression unit is to use a
standard electric motor and a speed-increasing gearbox. Such machinery is manufactured by
Atlas Copco, Dresser-Rand, Solar Turbines, MAN Turbo, etc. During the latest decades high-
speed machines have been pushed on the market as an interesting solution to increase the total
system efficiency and to minimise total costs.
20
Until the mid-1980’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 high-speed installations before
1990. Since the mid-1980’s, reliable electric adjustable speed control has been available for
medium voltage, megawatt-range, induction motors. As the acceptance of the induction motor
control technology in industry increased, it was only consequent that this technology was
considered to be applied also to high-speed use (Rama 1997).
1.2 High-speed machines
There are mainly two types of high-speed machines on the present-day market: High-speed
induction machines and high-speed synchronous machines with permanent magnet excitation.
However, minor research of claw-pole synchronous, synchronous reluctance and switched
reluctance high-speed machines is done as well. When the speed is high, centrifugal forces and
vibrations play an important role. Firstly, the rotor must have sufficient mechanical strength to
withstand centrifugal forces. Secondly, the designer must take the natural frequencies of the
construction into account. The critical frequencies may be handled in two ways; either the rotor
is driven under the first critical speed, which needs a strong construction and thick shafts, or the
rotor is driven between critical speeds. The latter obviously reduces the operating speed range
into a narrow speed area.
In induction machine applications - as far as the peripheral speed of the rotor is low enough, and
thus the mechanical loading is not a limiting factor - the laminated rotor with a squirrel-cage is
widely used. The first critical speed of this rotor type tends to be much lower than that of a solid
rotor. When the mechanical loading is heavy, solid-rotor constructions are used. Also in
permanent magnet rotors the laminated constructions with buried magnets can be used if the
mechanical stiffness of the shaft permits it. When the peripheral speed of a PMSM is high, a
solid steel rotor body is used and a magnet retaining ring or sleeve is needed. The retaining ring
is usually made of glass or carbon fibres, or of some non-ferromagnetic steel alloy material.
The issue of the state-of-the-art high-speed technology may be covered by making an analysis
of the articles dealing with the subject and an examination of the data sheets of the motor
manufacturers. Table 1.1 lists some high-speed electric machines that were selected from the
result of a literature search and table 1.2 gives some high-speed electric machine manufacturers.
The trend seems to be that for high-speed motors with power larger than 100 kW the induction
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 high-speed
applications in the megawatt range (Rama 1997) do exist and also small power applications
seem to be surprisingly general. Applications in the low voltage middle power range between
100 kW and 1000 kW and above 10000 min-1 are rarely used.
Table 1.1. Some high-speed electric machines selected from literature.
Power/kW Speed/ min-1
Motor type Reference:
41000 3750 Synchronous motor Rama (1997), gas compressor 38000 4200 Synchronous motor Kleiner (2001), gas compressor 13000 6400 Synchronous motor Steimer (1988), petrochem. application 11400 6500 Synchronous motor Lawrence (1988), gas compressor 10000 12000 Solid-rotor IM, caged Ahrens (2002), prototype 9660 8000 Induction motor Rama (1997), gas compressor 9000 5600 Synchronous motor Khan (1989), feed pump 6900 14700 Laminated-rotor IM McBride (2000), gas compressor 6000 10000 Laminated-rotor IM Gilon (1991), gas compressor 5220 5500 Solid-rotor IM, caged LaGrone (1992), gas compressor 2610 11000 Solid-rotor IM, caged Wood (1997), compressor 2300 15600 Solid-rotor IM, caged Odegard (1996), gas compressor 2265 12000 Induction motor Rama (1997), pump 2000 20000 Induction motor Graham (1993), gas compressor 1700 6400 Induction motor Mertens (2000), chemical compressor 270 16200 Laminated-rotor IM Joksimovic (2004), compressor 250 8400 Solid-rotor IM, end
rings Huppunen (1998a), blower
200 12000 Solid-rotor IM, caged Ikeda (1990), prototype 131 70000 Permanent magnet SM Bae (2003), micro-turbine 110 70000 Permanent magnet SM Aglen (2003), micro-turbine 65 30500 Coated
Solid rotor IM, caged Laminated-rotor IM
Lähteenmäki (2002), prototypes
62 100000 Coated solid-rotor IM Jokinen (1997), prototype 60 60000 Coated solid-rotor IM Lähteenmäki (2002), prototype 45 92500 Induction Motor Mekhiche (1999), turbo-charger 40 40000 Permanent magnet SM Binder (2004), prototype 30 24000 Permanent magnet SM Lu (2000), prototype
22
22 47000 Permanent magnet SM Mekhiche (1999), air condition 21 47000 Laminated rotor IM Soong (2000), cooling compressor 18 12
13500 13500
Solid-rotor IM, caged Solid-rotor IM Solid, slitted-rotor IM
Pyrhönen (1996), milling machine
11 56500 Laminated Kim (2001), compressor Table 1.2. High-speed stand-alone electric motor manufacturers in the power range over 100 kW.
Power range/kW Speed range/ min-1 Rotor type Manufacturer
1000 – 25000 6000 – 18800 Induction Alstom 30 – 1500 20000 – 90000 Claw Poles Alstom 500 – 20000 3600 – 20000 Induction ASIRobicon 100 – 1500 6000 – 15000 Induction Rotatek Finland 100 – 730 3600 – 14000 Induction ABB 100 – 400 3600 – 9000 Induction Schorch 40 – 400 10000 – 70000 Permanent magnet S2M 50 – 2000 20000 – 50000 Permanent magnet Calnetix 20 – 450 5500 – 40000 Permanent magnet Reuland Electric 3.7 – 100 3000 – 12000 Induction Siemens 1 – 150 – 25000 Switched reluctance SR Drives 1 – 20 – 15000 Switched reluctance Rocky Mountain Inc.
1.3 Solid-rotor constructions in high-speed induction machines
In the induction motor, in order to produce an electromagnetic torque Tem, and a corresponding
electric output 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 per-unit slip, which is defined
as:
S
sl
S
RS
S
sl
S
RS
S
sl
S
RS
ff
fff
ΩΩ
ΩΩΩs =
−==
−==
−=
ωω
ωωω . (1.1)
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
emR
emR
emReπ2 TpfT
pTΩP ===
ω (1.2)
The slip frequency fsl and the slip angular speed ωsl in the rotor are of great importance,
especially in solid-rotor machines since the slip angular speed, for instance, has a significant
role in determining the magnetic flux penetration in the rotor. The slip angular speed is one of
the factors determining the torque produced by the rotor. The I2R losses, however, in the rotor
depend on the per-unit slip s. For the design of a high-efficiency solid-rotor machine, one of the
design targets should be the minimisation of the per-unit slip.
Solid-rotor induction motors are built with a rotor core made of a solid single piece of
ferromagnetic material. The simplest solid rotor is, in fact, a smooth steel cylinder. The
electromagnetic properties of such a rotor are, however, quite poor, as, e.g., the slip of the rotor
tends to be large, and thus several modifications of the solid rotor may be listed. A common
property of the rotors called solid rotors is the solid core material that, in all cases, forms at least
partly the electric and magnetic circuits of the rotor. The first performance improvement in a
solid rotor is achieved by slitting the cross section of the rotor in such a way that a better flux
penetration into the rotor will be enabled. The second enhancement is achieved by welding
well-conducting non-magnetic short-circuit rings at the end faces of the rotor. The ultimate
enhancement of a solid rotor is achieved by equipping the rotor with a proper squirrel cage. In
all these enhancements the rotor ruggedness is best maintained by welding all the extra parts to
the solid-rotor core. Smooth solid-steel rotors may also be coated by a well-conducting
material. Five different basic variants of solid-rotor constructions are schematically shown in
Fig. 1.2.
The smooth solid rotor is the simplest alternative and thus the easiest and the cheapest to
manufacture. It also has the best mechanical and fluid dynamical properties, but it has the
poorest electrical properties. In practice, the manufacturing of a smooth solid rotor is not
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 copper-squirrel-cage solid
rotor gives three to four times as much torque as the same motor with a smooth solid rotor. The
fundamental rotor losses in a copper-cage solid rotor are only a fraction of those of a smooth
solid rotor. In addition, a squirrel-cage rotor construction gives a clearly better power factor –
comparable to the power factor of a standard induction motor – than a smooth rotor one.
The solid-rotor induction motor construction offers several advantages:
• High mechanical integrity, rigidity, and durability. The solid rotor is the most stable
and of all rotor types it maintains best its balance.
• High thermal durability.
• Simple to protect against aggressive chemicals.
• High reliability.
• Simple construction, easy and cheap to manufacture.
• Very easy to scale at large power and speed ranges.
• Low level of noise and vibrations (if smooth surface).
25
a)
b)
c )
d )
e )
Fig. 1.2. Solid-rotor constructions: a) smooth solid rotor, b) slitted solid rotor, c) slitted solid rotor with
end rings, d) squirrel-cage solid rotor, e) coated smooth solid rotor. Gieras (1995)
26
On the other hand, a solid-rotor induction motor has a lower output power, efficiency, and
power factor than a laminated rotor cage induction motor of the same size, which are
disadvantages that are mainly caused by the high and largely inductive impedance of the solid
rotor. The solid rotor impedance and its inductive part can be diminished in one of the
following ways:
1. The solid rotor may be constructed of a ferromagnetic material with the ratio of
magnetic permeability to electric conductivity as small as possible.
2. Using axial slits to improve the magnetic flux penetration to the solid ferromagnetic
rotor material.
3. A layered structure in the radial direction of the rotor may be made of appropriate
ferromagnetic and non-ferromagnetic high-conductivity materials (coated rotor).
4. A layered structure in the axial direction of the rotor may be made of appropriate
ferromagnetic and non-ferromagnetic high-conductivity materials (end-ring structure).
5. Use of a squirrel cage embedded in the solid ferromagnetic rotor core material.
6. The effects of the high impedance may be offset by the use of an optimum control
system.
7. Use the solid rotor in high-speed applications when the per-unit slip is low. The higher
the motor rotating frequency is the less important the rotor impedance will be. For
example: The rotor needs a 2 Hz absolute slip to produce the needed torque. If the
motor rotating frequency is 50 Hz the per-unit slip is 4 %, which means that 4 % of the
air-gap power is lost in the rotor copper losses. If the rotating frequency is 200 Hz the
same absolute slip results in a 1 % per-unit slip and, correspondingly, in a 1% per unit
rotor copper losses.
Solid-rotor induction motors can be used as:
• High-speed motors and generators.
• Two- or three-phase motors and generators for heavy duty, fluctuating loads,
reversible operating, and so forth.
• High-reliability motors and generators operating under conditions of high temperature,
high acceleration, active chemicals, and so on.
• Auxiliary motors for starting turbo-alternators.
• Flywheel applications.
27
• Integrated machines. The rotating part of the load machinery forms the rotor, for
example conveyer idle, where the stator can be outside or inside of the rotor.
• Eddy-current couplings and brakes.
1.4 Objectives of the work
The problem of calculating the magnetic fields in solid rotors has been a subject of intensive
study from the 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 solid-rotor IM are not good enough.
Since the use of high-speed machines became more popular from the beginning of the 1990’s a
few FEM studies about solid-rotor IMs have been published, but still the activities remained
low in this specified field.
The present study is done to establish a fast practical method for the design purposes
determined by the manufacturer of solid-rotor motors. The research has seven main objectives.
1) To create an analytical, multi-layer transfer-matrix method (MLTM method) based
calculation procedure for a slitted solid rotor equipped with copper end rings in order to enable
an accurate enough estimation of the behaviour of the electromagnetic fields in the slitted solid
rotor. When the field problem is solved the motor air-gap power is found by integrating the
Poynting vector over the rotor surface. The rotor behaviour is then connected to the traditional
equivalent circuit behaviour of the induction motor. 2) To introduce an analytical procedure by
means of which it is possible to precisely enough determine the losses of the solid-rotor IM. 3)
To find the best length to diameter ratio for a copper end ring slitted solid rotor. 4) To find the
best possible practical slitting patterns for the industrial motor solid rotor with copper end rings,
5) to introduce the power-dependent utilization factors for different types of solid rotors based
on the practical research results reached at LUT, 6) to compare the analytically found
electromagnetic results with the Finite Element Method (FEM) based solutions, and 7) acquire a
practical proof for the given theories by making careful measurements with appropriate
prototypes. The output powers of the prototypes vary between 250 kW and 1000 kW as the
speeds of the prototypes vary between 8400 min-1 and 12000 min-1. The main dimensions of the
250 kW – 300 kW prototype machines are: a 200 mm air-gap diameter, a 280 mm stator stack
effective length.
28
This work strongly focuses on the electromagnetic phenomena of the solid-rotor machine,
irrespective of the fact that mechanic and thermodynamic studies are of essential importance,
especially as high-speed machines are concerned. Usually, in practice, all of these three
demanding scientific fields need their own specialists to solve the exacting challenges in the
different fields. For that reason, the need of limiting this study to the electromagnetic
phenomena should be acceptable.
1.5 Scientific contribution of the work
In summary, the main scientific contributions of the thesis are:
1. The further development of the well-known multi-layer transfer-matrix method to be
used, especially, for the calculation of high-speed slitted solid-rotor induction motors.
Improvement of the multi-layer transfer-matrix method was achieved by introducing
into the method a new end-effect factor and a new curvature factor for slitted solid-
rotors equipped with well-conducting end rings. The new factors are functions of the
slit depths.
2. Definition of the best possible practical slitting of solid rotors equipped with well-
conducting end rings for high-speed induction motors in the medium power range.
3. Definition of the best possible rotor active length to diameter ratio for slitted solid-rotor
induction motors with well-conducting end rings.
4. Introducing of the power-dependent utilization factors for different types of solid
rotors.
5. Introduction of a new method to reduce the permeance harmonic content in the air-gap
flux density distribution by means of a new geometrical modification of a semi-
magnetic slot wedge. The slot wedge is formed as a magnetic lens.
Apart from these scientifically new contributions, the thesis also contributes, especially to the
practical engineer, in a valuable way, which may be summarized to be the following:
29
1. An analytical electromagnetic – and accurate enough - analysis of the solid-rotor
induction machine is introduced. The method is very useful in every-day practical
electrical engineering.
2. Discussion on the analysis of the analytical harmonic power loss calculation in solid
rotors. Methods of minimizing the harmonic power loss in the rotor surface are also
widely discussed.
3. New practical information on selecting the flux densities in the different parts of solid-
rotor induction machines in the medium speed and power range.
4. Some measures of diminishing the time harmonics caused by the frequency converter
are briefly introduced.
Several end-effect factors are presented in the literature on the subject. Usually, these factors
are introduced for a smooth solid rotor. They are based on the calculation of the penetration
depth, and should thus be a function of the rotor slip frequency. In practice, in a deeply slitted
solid rotor with well-conducting end rings, the axial rotor currents penetrate as deep as the slits
are. And, in practice, this current penetration depth is not depending on the slip when a normal
slip range of not more than a few percents is used. It is thus possible to use the real dimensions
of the end rings in the end-ring impedance calculations. The analysis assumes also that the
inductance of the end ring is negligible compared to the inductance of the slitted part of the
rotor.
Furthermore, a new curvature factor is defined for slitted solid rotors to be used in the MLTM
method when rectangular coordinates are used.
Slitting patterns for solid rotors have been studied earlier, but the examinations were in different
ways restricted; they were not done for high-speed machines, the parameter variation was done
within a very narrow range, the electromagnetically best slitting alternatives could be found but
the practical manufacturing conditions were disregarded.
According to the knowledge of the author, the utilization factors introduced in this thesis for
different types of solid-rotor induction motors have not been presented earlier. However, the
utilization factors for copper-coated solid-rotor induction motors were presented by Gieras
(1995).
30
1.6 Outline of the work
The multi-layer transfer-matrix method for a solid rotor was introduced by Greig (1967). Later,
several authors have used this method. The substitute parameters for a slitted solid rotor were
introduced by Freeman (1968). These form the basics for the calculation procedure introduced
here. In the second chapter, the history of the field calculation problem in the solid rotor is
discussed. The MLTM principles are repeated in chapter two.
Loss calculation of the solid-rotor IM is also one of the main objectives. When a solid rotor is
used extra attention must be paid to the eddy currents on the surface of the rotor solid steel.
Eddy currents are caused by the spatial and time harmonics of the air-gap magnetomotive force
(mmf) and the permeance harmonics as well. This is discussed in chapter three.
In chapter four the slitted solid rotor is examined and the MLTM and FEM calculation results
are compared. Also the measured results are given.
The conclusions of the research are given in chapter five.
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 solid-steel rotors. Since the
conventional induction machine theory proved to be inadequate for solid-rotor machines, the
need has grown to improve the methods of investigation. It has become necessary to determine
the solid-rotor machine performance directly based on the analysis of the electromagnetic
fields. The specific problems such as saturation, the effect of the finite axial length and rotor
curvature also affect the performance of the motor greatly and are, for this reason, of most
significant importance. In this study some of the known methods are combined and further
investigated in order to find a solution, which, in an appropriate way, gives consideration to all
the important rotor phenomena.
Although a smooth solid rotor is an extremely simple construction, the calculation of its
magnetic and electric fields is a demanding process because the rotor material is magnetically
non-linear and the electromagnetic fields are three-dimensional. Thus, to solve the solid-rotor
magnetic and electric fields fast and accurately enough is a demanding task. In the conventional
laminated squirrel-cage rotor induction motor design the magnetic and electric circuits can be
assumed to be separated from each other in the stator as well as in the rotor so that the electric
circuit flows through the coils and the magnetic circuit flows mainly through the steel parts and
the air-gap of the machine. For this reason, these phenomena can be examined separately.
Furthermore, in a traditional induction motor the magnetic circuit is made of laminated electric
sheets and end rings are included in the squirrel cage, and thus, without losing accuracy, it has
been possible to perform the examination in two dimensions and the non-dominant end effects
could be studied separately. In a solid rotor the steel material forms a path for the magnetic flux
and for the electric current, and, therefore, three-dimensional effects and non-linearity have to
be taken into consideration. Hence, the standard linear methods of analysis in which only
lumped parameters are considered, are no longer valid.
The rotor field solution could be solved by the three-dimensional FEM calculation, but it takes
far too much time to be used in every-day motor design proceeding. Besides, the modelling of a
rotation movement even more complicates the FEM calculation. Therefore, a three-dimensional
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 BH-curve. The assumption of an infinitely long rotor brings as a result a two-
dimensional analysis, but to achieve a good accuracy the end effects should be taken into
consideration. On the presumption of the rotor material being magnetically linear, a constant
value of 45° is given to the phase angle of the rotor impedance. The constant phase angle is
contrary to many experimental results, which have shown that the phase angle of non-laminated
steel rotors is far less than 45°.
An important feature of the solid-rotor induction machine is that the magnetic field strength at
the surface levels of the rotor is usually sufficient enough to drive the rotor steel deep into the
magnetic saturation. The limiting non-linear theory of the flux penetration into the solid-rotor
material considers that the flux density within the material may exist only at a magnitude to a
saturation level. This theory was used by MacLean (1954), McConnell (1955), Agarwal (1959),
Kesavamurthy (1959), Wood (1960d), Angst (1962), Jamieson (1968a), Rajagopalan (1969),
Yee (1972), Liese (1977) and Riepe (1981a). This rectangular approximation to the BH-curve is
good only at very high levels of magnetisation. This analysis gives a constant value of 26.6° 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 non-linear theory produce a constant power factor for the rotor impedance
independent of the rotor slip, material and current. That is, however, contrary to the
experimental results. In practice, the phase angle of the rotor impedance is somewhere between
these two extremes given by the linear theory and the limiting non-linear theory. Usually,
magnetic material saturation is a disadvantage that complicates the phenomena and decreases
the performance. It could, however, be determined that the saturation effects of the solid-rotor
steel, in this particular case, are beneficial since they increase the solid-rotor power factor. The
equivalent circuit approach was used by McConnell (1953), Wood (1960a), Angst (1962),
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 non-linear function for the BH-curve must be used.
The non-linear variation of the fundamental B1-H –curve is included in its entirety by
substituting the equation B1=cH(1-2/n), where c and n are constants. This fits the magnetisation
curve well. This form was used by Pillai (1969). He concluded that the rotor impedance phase
angle varies according to the exponent 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 transfer-matrix technique – for
determining the magnetic and electric field strengths and the current density in plane
conducting metal plates of constant permeability produced by an external impressed alternating
magnetic field. This method was later generalised by Greig (1967). Greig calculated the
electromagnetic travelling fields in electric machines. The generalised structure comprises a
number of laminar regions of infinite extent in the plane of lamination and of arbitrary
thickness. The travelling field is produced by an applied current sheet at the interface between
two layers. It is distributed sinusoidally along the plane of the lamination and flowing normally
to the direction of the motion. The transfer matrix calculates the magnetic and electric field
strengths of the following plane from the values of the previous plane using prevailing material
constants. The method is called multi-layer transfer-matrix method (MLTM method).
The MLTM method divides the rotor into a large number of regions of infinite extent. The
original MLTM method does not consider the rotor curvature, material non-isotropy or the end
effects, but the method gives consideration to the non-linearity of the material, because the
permeability and the conductivity of the rotor material are presumed to be constants in each and
every region separately. The tangential magnetic field strength and the normal magnetic flux
density will be calculated in every region boundary using the suppositions mentioned earlier.
After that the permeability and the conductivity in each region have been defined and hundreds
of regions have been calculated, it is possible to achieve very accurate results. (Pyrhönen
1991a).
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 Fourier-series. 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 air-gap does not vary in the radial direction. The
mistake made here is negligible when the air-gap is small compared to the diameter of the
rotor.
In the applied method a coordinate system fixed with the rotor is used, as it is shown in Fig. 2.1.
The origin is at the surface of the rotor and axially at its midpoint. The z-axis is taken in the
axial direction. The y-axis is normal to the rotor surface and the x-axis is in the tangential
direction, i.e. it is in circumferential direction. When the rotor is rotating at a slip s in the
direction of the negative x-axis, its position in the stator coordinates can be written as
prtsxx s
RS )1( ω−−= , (2.1)
35
where p is number of pole pairs, r is rotor radius, t is time and ωs is stator angular speed.
y
xz
Fig. 2.1. Coordinate system at the surface of the rotor.
The next abbreviation is taken into use. The constant a is dependent on the dimensions of the
machine
p
aτπ
= , where τ p is pole pitch, pD
p 2π
=τ . (2.2)
Equation (2.1) can be rewritten now
tsaxtax sR
sS 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∂
∂ DJH +=×∇ , ( 2.4)
36
t∂
∂ BE −=×∇ , (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 quasi-static, i.e. Maxwell’s displacement current is
negligible compared with the conducting current at frequencies which are studied in solid-rotor
materials, see App. C.
In addition, the material equations are needed:
ED 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 two-dimensional eddy-current 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
37
0=⋅∇ A . (2.12)
When equation (2.11) is substituted to Faraday’s law equation (2.5) we get
0=
∂∂
+×∇ At
E . (2.13)
The sentence in parenthesis has no curl and may thus be written as a gradient of a scalar
function −φ. Now, the electric field strength can be written in the following form
φ∂∂
∇−−=t
AE . (2.14)
The charge density ρ can be assumed to be negligible in well-conducting solid-rotor material.
Therefore, the divergence of the electric field strength is zero. The reduced scalar potential φ
describes the non-rotational part of the electric field strength. The non-rotational part is due to
electric charges and polarisation of dielectric materials. However, in a two-dimensional eddy-
current problem the reduced scalar potential must equal zero, see App. D.
Using equations (2.9), (2.10), (2.11) and (2.14) and keeping permeability µ and conductivity σ
as constants, equation (2.4) can be written
t∂
∂−=∇−⋅∇∇=×∇×∇
AAAA 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
AzA
yA
xA 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 time-harmonic solution may be found in the analysis. The vector
potential A is considered. It can be expressed in a time-harmonic form by
38
[ ]tszyxtzyx sje),,(Re),,,( ωAA = , (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 e jax and as a function of time t with the term
e j ss tω . The vector potential is obtained in form of a complex vector function
[ ])(j se),(Re),,,( tsaxzytzyx ω+= AA . (2.18)
Now, equation (2.16) can be written as a complex exponent function
iii Aa
zA
yA
)( 222
2
2
2
λ∂
∂∂∂
+=+ , (2.19)
where p
sj2
jd
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 air-gap length we get
r
2
δµλγ += 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.
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 xy-plane.
In addition, the depth of the penetration is assumed to be much smaller than the pole pitch.
The simplified equations in closed form for the vector potential in the x, y and z-direction are:
(Pyrhönen 1991a)
)(j se)
2sinh(
)sinh()ee()
2sinh(
)sinh(e tsaxyayy
x Lz
Lz
GA ωλλ
λ
λ
γ
γ +
−+= , (2.23)
)(j se)
2sinh(
)sinh()ee(j tsaxayyy L
zGA ωλ
λ
λ +−= , (2.24)
)(j se)
2sinh(
)cosh()
2coth()
2coth(ej tsaxy
z LzaLaLGA ωλ
γ
γγ
γγ
λ +
−+= , (2.25)
where
++
−=
)2
coth()2
coth()(
ˆj
r
2
00S
LaLa
KIG
γγ
λµλδ
µ, Na
pmK ξπ0 = . (2.26)
In the rotor the magnetic flux density equations are:
( ) )(j se)
2sinh(
)cosh(ee)
2sinh(
)cosh()
2coth()
2coth(ej tsaxyayy
x Lz
LzaLaLGB ωλλ
λ
λ
γ
γγ
γγ
λλ +
−+
−+= ,
(2.27)
40
( ) )(j se)
2sinh(
)cosh(ee)
2sinh(
)cosh()
2coth()
2coth(e tsaxyayy
y Lz
aLza
aLaLGaB ωλλ
λ
λλ
γ
γ
γ
γγ
γλ +
−+
−++= ,
(2.28)
)(j se)
2sinh(
)sinh(
)2
sinh(
)sinh(e tsaxy
z Lz
Lz
GB ωλ
λ
λ
γ
γλ +
−−= . (2.29)
The tangential and the axial magnetic flux components per unit width on the surface of the rotor
are found by integrating the respective flux densities:
)(j0
se)
2sinh(
)cosh(1)
2sinh(
)cosh()
2coth()
2coth(jd tsax
xx Lz
aLzaLaLGyBΦ ω
λ
λλ
γ
γγ
γγ
λ +
∞−
−+−+== ∫ , (2.30)
)(j0
se)
2sinh(
)sinh(
)2
sinh(
)sinh(d tsax
zz Lz
Lz
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.
41
AZ
Ax
HZ
ΦZ
Hx
Φx
1
0L / 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 per-unit length.
2.2 Calculation of a saturated solid-rotor
The electromagnetic fields in saturated rotor material can be solved with the MLTM method,
where the rotor is divided into regions of infinite extent. Fig. 2.3 describes the multi-layer
model and the coordinates used, Greig (1967).
In general, the current sheet
)(j se'Re taxJJ 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.
42
K B
H
ρ µK K K-1
K-1K-1 B
H
ρ µK-1 K-1 K-2
K-2
B
H k+1k+1 B
H
ρ µk
kk B
H
ρ µk k k-1
k-1 ρ µk-1 k-1
y
x
z
H -J'k
3B
H
ρ µ3 3
2
22 B
H
ρ µ2 2 1
11 ρ µ1 1
.
.
.
.
.
y = g K-1
y = g 1
y = g 2
k-1
k+1k+1
k+1
.
Fig. 2.3. Original two-dimensional multi-layer model (Greig 1967).
A stationary reference frame is chosen in which the exciting field travels with velocity 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,k-1, and the tangential component of the magnetic
field strength is Hx,k-1. The corresponding values on the upper boundary are By,k and Hx,k,
respectively (Greig 1967).
It is assumed that the regions may be considered planar, all end effects are neglected, as it has
been done for the magnetic saturation too; also the displacement currents in the conducting
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 µk+1 k+1 k
kregion k B
H
ρ µk k k-1
k-1
y = gk
region k - 1y = gk-1
d kωk
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
kyk
kx
ky
kkkkk
kk
k
kk
kx
ky
HB
HB
dd
ddHB
TΥΥβ
Υβ
Υ, (2.33)
where k
kk a µµ
Υβ
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,,
1e ygKyKy
KKBB −−
−= Υ , (2.36)
44
)(1,,
1e ygKxKKx
KKHH −−
−−= Υβ . (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,
11e gyyy BB −= Υ , (2.39)
)(1,1,
11e gyxx 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
ykk
kx
ky
HB
HB
TTT L , (2.41)
[ ][ ] [ ]
−
=
+−−
−
−
',
,121
1,
1,
JHB
HB
kx
kykKK
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 inter-layer boundaries up to the surface of the rotor. At this interface
Hx,k corresponds to the total rotor current. This rotor model may be combined with a
conventional equivalent circuit representation of the air-gap and the stator. Iterative adjustment
of Hx,1 is made to adapt the conditions at the rotor surface.
As By,k and Hx,k are resolved at all inter-region boundaries, it is then a simple matter to calculate
the power entering a region. The Poynting vector in the complex plane is
.*,, kxkzk HES = (2.43)
45
The time-average power density in (W/m2) passing through a surface downwards at gk may be
found by using the following expression: (Freeman 1968)
*,,,in Re5.0 kxkzk HEP −= , where k= 1, 2, .. K. (2.44)
Ez,k is the component of the electric field strength in the z-direction and it may be written as:
kyk
kz Ba
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
2Re kxkykxkyk HBHB
aP 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 I2R elaborated by the region is (Greig 1967)
kkkk PsPP =− ,mech . (2.48)
2.2.1 Definition of the fundamental permeability in a non-linear material
In a saturable material sinusoidally varying magnetic field strength creates a non-sinusoidal
magnetic flux density (Bergmann 1982). The amplitude spectrum of this flux density can be
numerically defined with the DC-magnetisation curve of the material. Fig. 2.5 shows how the
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 1B of the Fourier series of the flux density is obtained by a numerical
integration:
46
∫=π
01 )(d)sin()(
π2ˆ tttBB ωωω . (2.49)
The fundamental permeability of a particular working point is defined as
HBH ˆˆ
)ˆ( 11 =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 B1-H curve with DC-
magnetizing curve.
2.2.2 Rotor impedance
The rotor fundamental magnetomotive force in the air-gap, referred to the stator, is
a
HxHI
pNm xax
x
pj
2de'
2π42 R
0j
RR1 === ∫−τ
ξΘ , (2.51)
47
from which the rotor current referred to the stator is found:
xHNam
pI RR 2jπ'
ξ−
= . (2.52)
The air-gap flux of the machine is obtained by integrating the radial flux density at the rotor
surface over a pole pitch. Faraday’s induction law gives an equation for the rotor voltage per
phase referred to the stator:
yax
y Ba
LNxLBNU
p
p
Rs
2
2-
jRsR 2
2jde2
j' ξωξω
τ
τ
−=−= ∫ . (2.53)
Finally, the rotor impedance referred to the stator is found:
x
y
HB
pmLN
IUZ
R
R2
s
R
RR π
)(2''' ξω
== . (2.54)
2.3 Effects of axial slits in a solid rotor
The performance of an induction machine with a solid-steel rotor can be considerably improved
by slitting the rotor axially. The presence of slits has a significant influence on the eddy current
distribution in the rotor; the slits usher the eddy currents to favourable paths as the torque is
considered. The non-isotropy of the rotor body resulting from the slitting is in contradiction
with the boundary condition of the MLTM method. Thus, the analysis of the rotor fields is now
essentially a three-dimensional problem the solving of which, as the slitted nature of the rotor
surface is to be taken into account, is an extremely complex and laborious task. Slitted rotor
fields were studied by Dorairaj (1967a), Freeman (1968), Jamieson (1968b), Rajagopalan
(1969), Yamada (1970), Bergmann (1982), Jinning (1987) and Zaim (1999).
Jinning (1987) studied optimal rotor slitting. According to his calculation results, the optimal
number of slits is between 5 and 15 per pole pair. The optimal depth of a slit equals
approximately the magnetic flux penetration depth and the ratio between the slit width and the
slit pitch is between 0.05 and 0.15. Zaim (1999) analysed a slitted solid-rotor induction motor
by means of a FEM program, but only a few rotor slit parameters are used. Also Laporte (1994)
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
tt ττ
µµ wwy += , (2.55)
tut
ut
µτµµwwx +
= , (2.56)
tuut
utu
ww ρρτρρρ
+= . (2.57)
wt wu
τu
y
x
z
Fig. 2.6. Slitted solid-rotor surface.
49
2.4 End effects of the finite length solid rotor
In the previous study the rotor was presumed to have an infinite length. Now, the effects of the
finite length are considered.
The problem of the end effects in solid rotors causes an indisputable difficulty. Several of the
authors earlier mentioned did not take these effects into consideration at all. Omitting the
problem may be justifiable if the rotor is equipped with thick end rings which have very low
impedance and which make the current paths nearly axial. However, this supposition is not
valid even in solid rotors with copper end rings because according to the experience of the
author, when a solid rotor with copper end rings is used and the end effects are not considered,
the calculated results give a 10 - 30 percent better torque at the given slip compared to
measured results. Kesavamurthy (1959) introduced an empirical factor to modify the value of
the rotor conductivity to incorporate the correction for the end effects. The author does not
explain how the empirical factor for the end effect correction is achieved. Russel (1958)
assumed that the rotor current density is confined in a thin shell around the rotor. Also
Rajagopalan (1969) used this assumption. Jamieson (1968a) introduced the analysis in which
the eddy currents are assumed to continue in the body of the rotor. He gives an equation for a
correction factor of the end effects. Wood (1960c) made in his analysis a certain approximation,
the validity of which is questioned. Angst (1962) proposed a complex factor that is applicable to
the effective rotor impedance. Deriving the factor involves the solution of the three-dimensional
field problem under constant permeability. Yee (1971), too, solves the three-dimensional field
problem under constant 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 high-conductivity end rings
If the solid rotor is equipped with end rings made of a high-conductivity 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 well-conducting 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
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
)
2πtanh(
π2
1
1
p
p LL τ
τα−
= . (2.58)
Further based on this, a general end-effect 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 end-effect correction factor in the following way,
2
R
R1
211er )tanh(4
21
++=
DpLkQQK , (2.60)
where )tanh()(1R
R1
R
R1 D
pLkpLDQ +−= . (2.61)
where erc
cer1 ρ
ρttk = , 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
51
rings are made of non-magnetic 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 end-effect 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 end-effect 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 cross-section area At the
DC resistance of the rotor tooth may be written as
tR
RtR A
LRσ
= . (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
erer
πQA
DRσ
= . (2.63)
When a tooth current is marked as IsR, the end-ring current is (Richter, 1954)
=
R
sRer πsin2
Qp
II . (2.64)
The currents cause copper losses in a rotor
)2( 2erer
2sRtRRRCu, IRIRQP += . (2.65)
In a two-pole rotor the number of phases is equal to the number of teeth, thus the resistance of
the rotor phase is
52
+=
R
2
ertRR πsin2
Qp
RRR . (2.66)
The end-effect factor is defined as a ratio between the resistance of the rotor tooth RtR and the
resistance of the rotor phase RR:
R
sRer R
RK = . (2.67)
The described method sets the values for the end-effect factor between [0.5 … 0.7] when a
copper squirrel cage is used and between [0.7 … 0.9] for a solid-steel rotor with copper end
rings. These values indicate that even when a solid-steel 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 well-conducting 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
er2
2coth
2coth
2
1)(
γ
γγ
λL
aLaLaLsK
−
+
+= . (2.68)
This factor takes also the loading into account. Ker(s) is analogous to the end-effect factor
derived by Angst (1962). Furthermore, Yee (1971) declares that arg (Ker) is found to be very
small, thus, for typical solid-rotor machines, Ker can be simplified to a real constant. Except for
very small slip values, coth (λL/2) ≈ 1. Setting, in addition, γ = a,
53
2
2coth1
2coth1
er
−
+
+
=LaaL
LaaLK . (2.69)
Another theory proposed for the calculating of the end effects in a finite-length 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 air-gap and follows a circumferential path near the air-gap. Flux Φ2
enters the rotor at the air-gap and follows an axial path near the air-gap 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 two-pole rotor. a) Φ1 corresponds to the axial eddy currents and b)
Φ2 to the end currents.
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 multi-layer transfer-matrix method.
Using equation (2.1) the equations may be expressed with respect to the stator coordinates. The
x-coordinate in stator reference frame is marked as x1.An annotation H0 is used.
λµ
GH 10 = . (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()
2coth()
2coth(ee )(j4
π3j
00s1
LzaLaLHH tax
yx
γ
γγ
γγ
λω , (2.71)
−−= +=
)2
sinh(
)sinh(
)2
sinh(
)sinh(ee )(j4
πj
00s1
Lz
Lz
HH taxyz
λ
λ
γ
γω , (2.72)
)2
coth(eee )(j4π3j
02
s1 LHH aytaxLzx λω+
== , (2.73)
aytaxLzy HH eee )(j4
πj
02
s1 ω+
== . (2.74)
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)
)
2sinh(
)sinh(e/e )(j
lin0s2πj
00s1
Lz
sHE taxyx
γ
γρµµω ω+
=−= , (2.75)
)(jlin0s
2πj
02
s1ee/e taxayLzx sHE ωρµµω +
=−= , (2.76)
)(jlin0s0
2
s1ee/ taxayLzy sHE ωρµµω +
=−= . (2.77)
The saturated components Hx(y=0) and Ez(y=0) are defined by the non-linear MLTM method,
equation (2.42), when the electric field strength in z-direction at the surface of the rotor
according to equation (2.45) is
Rs
0 yyz Ba
sE 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
Bz
Ex
E yxz
∂
∂=
∂∂
−∂
∂ . (2.79)
By choosing only the component that corresponds to the flux Φ2 equations (2.75) and (2.79)
give
∫ +=∂∂
=)
2sinh(
)cosh(e/ed )(j
02πj0 1
2 Lz
sjH
tzEB tax
linss
Φs
γ
γγρµµω
ωω . (2.80)
By integrating over the surface y = 0, the unsaturated path flux Φ2 is obtained as
56
sHa
xzBΦ
p
p
L
LΦ /4dd lin0s0
s
2
2
2
2
2 2ρµµω
ω
τ
τ
== ∫ ∫− −
. (2.81)
The air-gap voltage of the machine is calculated with Faraday’s induction law
)(2
j 21s ΦΦNU +−=ξωδ . (2.82)
By using the complex Poynting vector, see App. E, the average power density flow into the
surface can be defined as
*
21 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)(
21d)(
21
d)(21d)(
21π'
0
-
0
-2/
*2/2/
*
2/
2/
2/
2/
2/0
*00
*0
−+
+−=
∫ ∫
∫ ∫
∞ ∞====
− −====
yHEyHE
zHEzHED
LzyLzxLzxLzy
L
L
L
LyzyxyxyzS
(2.84)
In equation (2.84) the field variations in the direction of the x-coordinate 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).
57
2.5 Effect of the rotor curvature
The previously defined end-effect 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 y-direction, 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
RC,
21D
gDK kk
−−= . (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.
58
kk
k wwk
u,
u
u,
t,ty ''
')(
ττµµ += , (2.88)
tut,
u,tx '
')(
µτµ
µww
kk
k
+= , (2.89)
k
k
wwk
t,uut
u,tu
''
)(ρρτρρ
ρ+
= . (2.90)
g
Fig. 2.8. Effects of the curvature to the cross-section area of the tooth in slitted solid rotors.
The field calculation can also be executed in the polar coordinates by the multi-layer transfer-
matrix method, and thus the curvature effects are taken into account. The multi-layer model is
illustrated in Fig. 2.9.
KK-1
k+1
k
k-1
2
1
HK-1HK H k+1 Hk Hk-1 H 2 H 1
rr
r
rr
rK-1k+1
kk-1
2
1E
EE
EE
E
12
k-1k
k+1K-1
B K-1
B 2
B 1
B
B
B
k-1
k
k+1
r
z
Fig. 2.9. The cross-section through a K-layer cylindrical induction device.
59
The model is assumed to be infinitely long in the z-direction, so the end effects have to be taken
into consideration by the end-effect factor or the linear end-field calculation. Now, the transfer
matrix between each region k is according to Freeman (1974)
[ ]
=
=
−
−
−
−
1,
1,
1,
1,
,
,
kx
kyk
kx
ky
kk
kk
kx
ky
HB
HB
dbca
HB
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
dr
,
11
j−=β , (2.96)
kp
k
dr
,2
j=β , (2.97)
kr
kp,
,
µµ
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 multi-layer model of the rotor (Fig. 2.3) and it is commenced by assuming a low
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 inter-layer 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 air-gap and the stator. Iterative adjustment of H1 was then used to
attain a specified machine operation condition.
To take into account the non-linear magnetization characteristic of a solid-steel 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
BH-curve 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 non-isotropic 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 three-dimensional problem which must be taken
into account in the analysis. When well-conducting 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 end-ring 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 copper-end-ring solid rotors, hence the method described above has been used.
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).
62
3. On the losses in solid-rotor machines
The power losses in an electric motor determine the efficiency of the motor and also the cooling
that is required keeping the temperature below the upper limit. The motor torque determines
mainly the needed rotor size. Therefore, when speeds are used that are higher than those used in
conventional machines, it may be allowed to considerably reduce the size of the motor at the
same output power, if the high-speed machine has a better efficiency or if it is more effectively
cooled than the normal speed machine; the active mass of a 10000 min-1 motor can be 1/3 of
that of a 3000 min-1 motor. In order to be able to reduce the motor size, the motor efficiency
must be very high. If the motor output powers and the efficiencies are the same for the 3000
min-1 as for the 10000 min-1 motor the loss density in the high-speed version may be three times
as high as in the 3000 min-1 machine. This fact sets high demands to the design of the motor
cooling arrangement and, therefore, high-speed machines are often effectively ventilated
through the air-gap. In high-speed machines the effectiveness of the motor cooling and
especially the rotor cooling form the main limiting factors for the rated power of the machine to
be determined. If IP54, IC01 totally closed motor enclosures are demanded, the high speed
itself obviously brings no extra advantage with respect to the motor size. In such a case, 3000
min-1 and 10000 min-1 machines are about equal in physical size.
The power losses in an electric machine can be divided into mechanical and electrical losses.
The friction and cooling losses are included in the mechanical losses. The electrical losses are
put into two types, the fundamental frequency losses – which are core losses and winding ohmic
losses – and the harmonic losses. The harmonics in the air-gap magnetomotive force (mmf)
produce deviations in the mmf wave at higher frequencies than the fundamental frequency. The
air-gap harmonics can vary either in time or in space. The time-dependent harmonics are caused
by a non-sinusoidal power supply and the spatial harmonics are created by the machine discrete
mechanical structure. The harmonic deviations in the air-gap mmf wave cause losses especially
in the solid steel parts of the motor, since these harmonic waves penetrate into the conducting
material and cause eddy currents, which produce ohmic losses in the steel. In a solid-rotor
induction motor, the harmonics in the air-gap mmf are particularly detrimental, since in solid
steel the eddy currents have free and open paths to accrue. Harmonic losses, which are part of
the additional losses, constitute only 2 … 5 % of the total losses of laminated-rotor machines.
But, in solid-rotor machines the harmonic losses are typically about 10 % of the total losses, and
if the solid-rotor machine is not designed precisely, the portion of the harmonic losses can reach
63
up to 50 % of the total losses. This sets additional postulates in the design of the solid-rotor
induction motors.
This chapter mainly focuses on the harmonic eddy current losses in the surface area of the solid
rotor, since this rotor type differs significantly from the laminated rotor machine. One of the
aims of this thesis is to create a calculation program, which calculates the performance
characteristics of the solid-rotor induction motor. Hence, also the traditional calculation
methods of the stator steel and copper losses are studied briefly.
3.1 Harmonic losses on the rotor surface
The discrete stator windings and the reduction in the magnetic flux density under the stator slot
opening cause harmonics in the air-gap flux density, although the supplied phase voltages were
pure sine waves. The harmonics caused by the discrete coil distribution in the periphery of the
stator yoke are called winding harmonics and the harmonics caused by the slot openings are
called permeance harmonics. If the winding harmonic and the permeance harmonic are of the
same order, the particular harmonic is called slot harmonic. The harmonics generate remarkable
losses in a solid-rotor surface. The effects of these harmonics in solid steel and in asynchronous
motors were examined by Gibbs (1947), Agarwal (1960), Stoll (1965), Bergmann (1982) and
Pyrhönen (1994). In the following these phenomena are examined more closely.
When designing a solid-rotor induction motor, it is extremely important to minimise the
deviation from the sine wave of the mmf on the surface of the rotor, since the deviations create
energetically eddy currents in the solid rotor. Experimental results show that even by using a
smooth high-speed solid rotor, fair motor properties may be achieved at quite a low relative
slip. As an example, Pyrhönen (1991a) reported nominal slips for smooth rotors of about 1 %. If
a rotor type producing more torque had been used, only a small improvement in the motor
power could have been reached without extra loss minimising methods. This demonstrates that,
if the stator losses and the harmonics in the air-gap are kept low, a high efficiency of the solid-
rotor induction motor can be reached.
3.1.1 Winding harmonics
The winding harmonics are the result of building up the winding of conductors with a finite
width in the form of turns concentrated into individual coils, and thus the air-gap mmf wave is
64
not sinusoidal but stepped as shown in Fig. 3.1. The symmetrical m-phase stator winding
creates mmf harmonics of order (Richter 1954)
,...3,2,1,0,12 ±±±=+= kkmν . (3.1)
For a three-phase AC-motor the harmonics are of the order:
ν = 1, −5, 7, −11, 13, −17, 19, −23, 25, −29, 31, −35, 37, −41, 43, −47, 49, etc.
The harmonics of positive order rotate in the same direction as the fundamental wave, and the
harmonics of negative order rotate in the opposite direction with respect to the fundamental
mmf wave. The harmonics induced voltages are included in the voltage of the fundamental
frequency. The number of pole pairs of the harmonic ν is νp and the angular velocity of the
harmonic with respect to the stator is
νωω ν
ss = . (3.2)
ΘΘ$
-1
-0,8
-0,6
-0,4
-0,2
0
0,2
0,4
0,6
0,8
1
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 491
Fig. 3.1. Waveform of the mmf of a 48-slot three-phase stator winding at the moment when iw = iv =
−1/2 iu. w/τp = 1.
65
The amplitude of the νth mmf expressed with the fundamental amplitude 1Θ)
and the winding
factors is
νξ
ξΘΘ νν
11
))= . (3.3)
On the other hand, the winding factor describes how the harmonic mmf and the magnetizing
inductance are connected together. For the winding where the q is an integer the winding factor
is
=
mqq
mw
p
2πsin
2πsin
2πsin
ν
ντ
ν
ξν , (3.4)
where w/τp is the winding pitch; w is the coil span and τp is the pole pitch. The slip of the rotor
with respect to the νth stator harmonic is
)1(1 ss −−= νν , (3.5)
and thus the angular velocity of the νth harmonic in the rotor is
νν ωω sSR = . (3.6)
The spatial harmonic waves thus move at a low speed in the air-gap and the rotor rotating at a
small fundamental slip has to pass them at high slips. The effects of the harmonics can be
studied with the help of the equivalent circuit given in Fig 3.2 (Agarwal 1960).
Considering the air-gap and stator phenomena, the magnetizing reactance of the νth harmonic
referred to the stator is (Agarwal 1960)
m12
2
1m
1 XXνξ
ξνν
= . (3.7)
66
RS
R'R1/s
R'R5/s5
R'Rν/sv
XSσ
X'Rσ1
X'Rσ5
X'Rσν
Xm1
Xm5
Xmνν = + = ± ± ±2 1 1 2 3km k, , , ,...
Fig. 3.2. Simplified complete equivalent circuit of the induction motor including the harmonic
machines.
The flux penetration into the conducting solid-rotor medium and the eddy current losses created
are considered next. For simplicity, it is assumed that the permeability and the conductivity are
constants. The Maxwell equations (2.4) - (2.7) give a differential equation to the magnetic field
strength at the surface of the rotor
HH σµωνR2 j=∇ . (3.8)
Because the pole pitch of the harmonic ν is small, the end effects may be ignored without
making a big mistake. Because the penetration depth of the harmonic νth is also small due to the
large slip, the curvature has no significant influence on the solution, and the problem is
regarded as a plane wave penetration into a conducting medium. The tangential magnetic field
strength is (Pyrhönen 1991a)
0j R2
2
=− ννν σµω
∂∂
xx H
yH
. (3.9)
67
The differential equation may be solved, since the magnetic field strength must vanish when y
goes towards minus infinity. The solution is
y
xx HHσµω
νν
νR2j1
0 e+
= . (3.10)
The axial electric field strength is
νννν σµωσ∂
∂σ xxz HH
yE R2
j111 +== . (3.11)
Since the surface current value corresponds to the surface magnetic field strength ( ss HnJ ×= ,
where n is the surface normal unit vector) the relation between the electric and magnetic field
strengths may be called surface impedance Zν of the harmonic ν:
σµωσ
νν
νν RR 2
j1+==
x
z
HE
Z . (3.12)
As the fundamental wave mmf and the rotor current were solved, it can be done also for the
harmonic wave. The harmonic mmf and the harmonic rotor current for the harmonic ν is
according to equation (2.51) and equation (2.52)
v
xvxaxvv
v
vv a
HxHI
pNmΘ
v
v
j2
de'22π
4 R0
jRR === ∫
−τ
ξ , (3.13)
νν
νν ξ
νaNm
pHI x
2jπ
' RR = , (3.14)
where Dpa ν
τνν
2π== . (3.15)
The harmonic ν induces a voltage, which is referred to the stator
68
∫−
−=−=2
2
Rsj
RsR 22jde
2j'
ν
ν
ν
τ
τν
ν
νν
νν
ξωξω yxa
y Ba
LNxLBNU . (3.16)
The normal flux density can be expressed by the axial electric field strength
νν
νν ω zy EaB R
RR −= . (3.17)
The rotor harmonic impedance can now be written
ν
ν
νν
νν ξ
τωω
x
z
p HE
NpmL
IUZ
R
R2
R
s
R
RR )(2
''' == . (3.18)
The impedance, which the stator current flows through, is the parallel connection of the
magnetizing impedance and the rotor impedance of the harmonic ν
νν
ννν
mR
mR
j'j''
XZXZZ
+= . (3.19)
The induced air-gap voltage of the harmonic ν is
νδν 'S ZIU = . (3.20)
The air-gap power of the harmonic ν is
⋅=
ν
δνδ
R
2
'Re3
ZU
P ν . (3.21)
Finally, the harmonic torque is obtained as
v
vv
PTsωδ
δ = . (3.22)
69
On the rotor surface several alternating excitations of different frequencies are superimposed.
The effects of the fundamental can reliably be calculated by determining the fundamental
permeability µ1, but excitation of the surface of the rotor varies in such a complicated manner
that great difficulties arise when trying to determine the permeabilities for the numerous
harmonics. Usually, just one incremental permeability µr is used for all harmonic mmf. For
example Bergmann (1982) has used the value µr = 40µ0 to describe the behaviour of all the
harmonics in a solid rotor (Pyrhönen 1991b). This value is also used in this study.
3.1.2 Permeance harmonics
The magnetic conductance Λ of the smooth air-gap is
δµ0=Λ , (3.23)
where δ is the air-gap length. A slotted stator and a smooth rotor surface are assumed.
According to Heller (1977), the harmonics caused by the stator slotting have an effect on the
air-gap permeance function, which may be written for a stator with QS slots
∑∞
=
−=1
S0 )cos()(ν
ν ανα QΛΛΛ , (3.24)
where δ
µ
CkΛ 0
0 = , (3.25)
where kC is Carter factor and the magnetic conductance of the harmonic ν is
=
u0 τδ
βµ ννoFΛ , (3.26)
where )π6.1sin(
2-0.78
5.0π41
u2
u
2
u
u τν
τν
τν
ντνo
o
ooF
+=
, (3.27)
70
and β is defined in equation (3.32). Let us first presume that a smooth magnetic pole is facing a
slotted armature. The air-gap flux density on a surface of the rotor is examined under one stator
slot pitch. If, without slotting, the value of the magnetic flux density in the air-gap were Bmax,
the flux density will now, with a slotted stator, drop to a value Bmin, which is a function of the
slot opening o and the air-gap length δ, Fig. 3.3. It should, however, be remembered that in the
case of a fast rotating solid rotor the rotor surface eddy currents mainly cancel out the flux
density dip under the slot opening. These eddy currents, naturally, create the permeance
harmonic losses on the rotor surface. In the case of a laminated rotor the flux density dip is
more real. At the same time, the mean magnetic flux density decreased from the original value
Bmax to the value Bave. Therefore, this change in the mean magnetic induction over the slot pitch
will correspond to a fictional increase of the air-gap of an un-slotted circumference from the
value δ to the value δ’.
Bmax
Bmin
BBn
o
o'
δ α
τu
B
Fig. 3.3. Distribution of the magnetic flux density on a surface of the rotor above a stator slot.
The relation between these quantities is expressed by the equation
δδ Ck=' , (3.28)
where kC is called Carter factor after F. W. Carter (1901). From that follows also the relation
71
maxC
ave1 Bk
B = . (3.29)
The Carter factor is determined as (Heller 1977)
γδτ
τ−
=u
uCk , (3.30)
where
δ
δδδδ
γ o
oooo
+
≈
+−
=
521ln
2arctan
2π4
2
2
. (3.31)
The amplitude of the magnetic flux density drop Bn at the axis of the slot is given by the relation
maxn 2 BB β= , (3.32)
where β is also a function of the ratio between the slot opening and the air-gap length. From
equation (3.32) we get
)1(2
2122 2
2
max
minmax
max
n
uuu
BBB
BB
+−+
=−
==β , (3.33)
where 2
21
2
++=
δδoou . (3.34)
The magnetic flux density varies in a bit wider distance than the slot opening. This effective slot
opening o’ is
βγδ
='o . (3.35)
Now, we obtain a new form for the Carter factor
72
γδτ
τβτ
τ−
=−
=u
u
u
uC 'o
k . (3.36)
The equations (3.32) - (3.36) are given by Richter (1967). With a bilateral slotting, i.e. the stator
and rotor slotting, the magnetic relations are very obscure and depend on the instantaneous
position of the two slot systems, so that the calculation of the resultant Carter factor may be
approximated as follows. The resultant Carter factor for two-sided slotting is
RC,C,SC kkk = , (3.37)
and we obtain the magnetic conduction in the form
)()()( r210
2,1 ααΛαΛµδαΛ −= , (3.38)
where αr is an angle of the rotor slotting displacement with respect to the origin of the stator
slotting.
If it is desired to accurately examine the air-gap permeance in a more complex slot opening
geometry, the Maxwell equations should be solved numerically. The best way to do this is to
use a finite-element method. Fig. 3.4 a) shows the finite element mesh in a vicinity of a stator
slot opening and Fig. 3.4 b) gives the magnetic flux lines in the same situation.
a) b) Fig. 3.4. a) Mesh plot in a vicinity of a stator slot opening. b) Magnetic flux lines under one stator slot.
73
Heller (1977) has introduced an equivalent relation for the magnetic flux density over the slot
pitch, when the origin is fixed to the middle point of a slot
=
<<
−−=
elsewhere,)(
6.10,6.1πcos1()(
max
max
BBD
oo
DBB
α
ααββα . (3.39)
In this equation it is assumed that the slot opening affects the magnetic flux density distribution
up to the distance ol ⋅= 8.0 from the centre of the slot. Fig. 3.5 shows the difference between
the solutions for a flux density distribution when calculated with FEM and with equation (3.39).
The results are very similar in Fig.3.5a, where the o/δ is 2.5. In Fig. 3.5b the o/δ is 1 and the
assumption of the distance of the slot opening is not valid anymore, but the minimum value of
the flux density is still accurate.
0,5
0,6
0,7
0,8
0,9
1
1,1
0 0,2 0,4 0,6 0,8 1x / τ u
B / B max
by FEM
by Heller (1977)
a)
0,88
0,9
0,92
0,94
0,96
0,98
1
1,02
0 0,2 0,4 0,6 0,8 1x / τ u
B / B max
by FEMby Heller (1977)
b) Fig. 3.5. Distribution of the magnetic flux density on a surface of the rotor above a stator slot pitch. a)
The ratio o/δ is 2.5, b) the ratio o/δ is 1.
Bergmann (1982) introduced Fourier series to evaluate the magnetic flux density over one slot
pitch
−−= ∑
∞
=1S1C,Save )cos()1(1)(
kk
k kQakBB αβα , (3.40)
where
−
=2
u
u1
)'(1π
)'πsin(
τ
τokk
oka k , (3.41)
74
Let us consider an arbitrary magnetomotive force, which is superposed on the permeance
function of the slotted air-gap. The resulting air-gap magnetomotive force in a three-phase
machine can be expressed with the sum of the individual harmonic mmfs (Bergmann 1982)
∑∞
−∞=+=
−=
gg 16
j00 ej)(
ν
νανΘαΘ . (3.42)
The place dependent permeance function corresponds to the inverse of the air-gap
+−= ∑
∞
=
−
1
jj1 )ee(1
'1
)(1
SS
k
kQkQkb αα
δαδ, (3.43)
where 11C1
1u))1(( ++−= Niqkkk akb β . (3.44)
Factor 11u))1(( ++− Niqk takes into consideration the position of a tooth or slot depending on the
number qu of the stator slots per phase and pole and the number iN1 by which the coil span
deviates from the pole pitch. The air-gap function can be written as
αν
νννν
ν
ανανανν
δµ
δµα
p
gg k
kqkqk
gg k
pkqpkqk
p
ΘΘbΘ
bΘB
j
16 1)6(0)6(010
0
16 1
)6(j)6(j1
j0
0
e)('2
j
)ee(e'2
j)(
uu
uu
−∞
−∞=+=
∞
=−+
∞
−∞=+=
∞
=
−−+−−
∑ ∑
∑ ∑
+−=
+−=
. (3.45)
With respect to the earlier statement we can calculate the induced stator voltage
∑ ∑∞
−∞=+=
∞
=−
−+
+
−+
+−=
gg k
kqkq
kqkq
k Ikq
Ikq
bIXU16 1
)6(muS
)6(S)6(m
uS
)6(S1mmS u
uu
u
66j
νν
ν
νν
ν
νννδ ν
νξ
ξν
νξ
ξ.
(3.46)
Here, Xmν is the magnetizing reactance of the νth harmonic motor. The annotation
kv
kqk akb 1SC,
S
)6(S1 2
1s βξ
ξ ν =± , (3.47)
75
which is obtained using equation (3.44) and the definition of the winding factor, simplifies
equation (3.46). Now, the voltage equation simplifies to the form
)(j m16
mmmS −
∞
∞
+∑ −−= k
g=-g+=
k IIIXU νν
νννδ , (3.48)
where )6(ms
1C,Sm u621
kqkk Ikq
akI ±± ±= νν ν
νβ . (3.49)
The equations (3.48) and (3.49) are valid for all kinds of three-phase single- and two-layer
windings. According to equation (3.48), the νth harmonic motor can be illustrated as an
equivalent circuit in Fig. 3.6. (Bergmann 1982).
The source currents of Fig. 3.6 are defined by equation (3.49), which shows that the
fundamental motor and the harmonic motors cannot be solved separately because they have an
effect on each other via the current source. The harmonic machines, the ordinal of which
deviate by ±6kqu, create the current sources of the harmonic motor ν. Bergmann (1982) has
shown that the power related with the current sources disappears, since every source power has
a counterpart that makes the power sum zero. Thus, the current sources do not disturb the power
balance of the machine.
This method, however, is quite inconvenient, because every harmonic machine is connected
with numerous other machines via the permeance function. No big mistake is made if the
method is simplified by leaving out all slot waves that are generated without the fundamental.
This can be done, because the winding factors of any other than the slot harmonics are small.
Slot harmonics do occur at ordinals
,...3,2,1,11S ±±±=+=+= kkqp
Qk pν (3.50)
and they have the same winding factor as the fundamental. If the method is simplified, the
voltages of the slot harmonics are
+−= +++ 1mu1C,S)61(m)61(m)61( )61(
21j
uuIkqakIXU kkqkqkq u
βδ . (3.51)
76
Z'RνXmν
Imνk+ Imνk-
IS I 'Rν
Imν
U 'Sδν
Fig. 3.6. Equivalent circuit for the νth harmonic motor. The current sources represent the effect of the
stator slots and they generate additional voltages in the magnetizing reactance.
3.1.3 Decreasing the effect of the air-gap harmonics
The disadvantages of the distributed stator winding can be reduced significantly, if the stator
slot number is increased. This will decrease the induced harmonic voltages. When a two-layer
short-pitch winding is used, the winding factors of the harmonics are decreased to a
considerable degree.
In Fig. 3.7 the winding factors of the full-pitch windings and the 5/6-short-pitch windings are
illustrated for both a two-pole stator and a four-pole stator, when the stator slot number is 48.
The 5/6-short-pitch winding decreases every other harmonic pair, influencing particularly the
5th and 7th harmonics. For a 5/6-short-pitch winding the number of slots has to be divisible
evenly by 2pm. Therefore, the possible numbers of the stator slots are for a two-pole machine
12, 24, 36, 48, 60, etc; and for a four-pole machine 24, 48, 72, etc.
The slot harmonics do occur at
,...3,2,1,1S ±±±=+= kp
Qkν , (3.52)
and they have the same winding factor as the fundamental. It should be mentioned that in a
four-pole machine the slot harmonics do occur twice as densely as in a two-pole machine, and
they appear at the first time at the order number that is half the stator slot number. Thus, a four-
pole machine has higher harmonic losses than a two-pole machine. The numerical values of the
77
winding harmonics repeat themselves between the slot harmonics as it can be seen by
comparing the values in the figure.
0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97ν
ξ
p=2, full-pitch windingp=2, 5/6-short-pitch windingp=1, full-pitch windingp=1, 5/6-short-pitch winding
p =2p =2p =1p =1
Fig. 3.7. Winding factors of full- and 5/6-short-pitch two- and four-pole three-phase 48-slot stator
windings.
In Fig. 3.8 the amplitude ratio of the harmonic magnetomotive forces compared to the
fundamental magnetomotive force of the full-pitch windings and the 5/6-short-pitch windings
are illustrated for both the two-pole stator and four-pole stator, when the stator slot number is
48.
The best winding pitch is 5/6, since it gives the minimum value for the leakage factor σδ, which
is defined by Richter (1967)
2
1 1∑
≠
=
ν
νδ νξ
ξσ . (3.53)
The leakage factor illustrates the harmonic content of the induced air-gap voltage. The
following flux distortion factor
78
∑≠
=
1 1ν
νδ νξ
ξβ , (3.54)
however, according to the author, illustrates the harmonic content of the magnetomotive force,
and, in that way, also the harmonic content of the magnetic flux density in the air-gap. In table
3.1 the harmonic contents of the magnetic flux densities in the air-gap are given for different
windings.
0,001
0,01
0,1
1
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97
ν
v ν / v 1
Q = 36, p = 1 full-pitch winding
Q = 48, p = 1 full-pitch winding
Q = 48, p = 1 5/6-short-pitch winding
Q = 48, p = 2 full-pitch winding
Q = 36, p = 1,
Q = 48, p = 1,Q = 48, p = 1,
Q = 48, p = 2,
Fig. 3.8. Amplitude ratio of the harmonic magnetomotive forces compared to the fundamental
magnetomotive force for different three-phase windings.
Table 3.1. Harmonic contents of the magnetic flux density for different stator windings.
Winding type Q = 36, full-pitch winding
Q = 48, full-pitch winding
Q = 48, 5/6-short-pitch winding
Flux distortion factor βδ, p = 1 23.5 % 19.9 % 12.2 %
Leakage factor σδ, p = 1 4.5 ‰ 3.5 ‰ 1.4 ‰
Flux distortion factor βδ, p = 2 40.6 % 34.8 % 24.6 %
Leakage factor σδ, p = 2 12.6 ‰ 8.1 ‰ 5.5 ‰
The winding harmonic losses (as well as the permeance harmonic losses) can also be reduced
by decreasing the effects of the air-gap harmonics on the conducting medium. By increasing the
air-gap length the winding harmonic effects on the rotor surface are reduced to a noticeable
degree. The eddy currents in a conducting medium can be decreased by using high resistivity
79
materials on the surface of the conducting medium. However, the design of the stator slot
opening is also of significant importance.
In solid-rotor induction motors the minimising of the permeance harmonics in the air-gap of the
motor is a very significant means of reducing the additional losses. The air-gap length has a
very important role when the flux harmonics on the surface of the rotor are studied. If the air-
gap length is increased, the flux distribution will smoothen on the rotor surface. Fig. 3.9 shows
how the flux density distribution becomes smoother when the air-gap length increases. Thus,
the air-gap length in the solid-rotor high-speed machine should be increased, compared to the
laminated rotor machine. A longer air-gap length increases the magnetizing current of the
motor, and thus the stator copper losses will be increased. Therefore, the loss minimum can be
found between the rising stator copper losses and the diminishing rotor harmonic eddy current
losses. The air-gap lengthening reduces the power factor as well. The flux distribution will also
be smoothened if the slot opening is narrowed. This expedient should be used, but the limit is
set by the winding manufacturing criteria.
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x / τ u
B / B max
1
2
3
4
Fig. 3.9. Flux density distribution under one slot pitch on the surface of the non-conducting rotor with
different ratios of a slot opening length and an air-gap length. The slot-opening length is kept
constant and is marked by dashed lines. The ratios o/δ are 1) 1, 2) 1.25, 3) 1.67 and 4) 2.5.
80
A longer air-gap length and narrower slot opening may not be enough in all cases. For this
reason, the stator slot opening has to be modified or a semi-magnetic wedge has to be used as a
slot lock component so that the permeance harmonics in the air-gap flux are minimised. To
solve the magnetic flux distribution on the surface of the rotor under a slot opening it is possible
to find the appropriate slot geometry. The traditional semi-closed slot opening is illustrated in
Fig. 3.4. In the slot opening geometry, which has a small, ¼-part of a circle, nodule on both
sides of the slot opening was found, Fig. 3.10 (Pyrhönen 1993). These extra pieces guide the
flux under the slot opening and thus reduce the flux dip depth. However, the nodules have some
disadvantages. They are of a considerably small size and thereof hard and expensive to
manufacture into the stator laminates. Problems have occurred during the manufacture
concerning e.g. the selecting of proper tools and also the durability of the tools. The spacing
between the nodules is made very narrow, which makes the provision of coiling to the stator
much more difficult. Therefore, the solution is not considered to be the most optimal in all
cases. The larger the machine the easier the stator of this kind is to manufacture. The suitable
minimum measurements for the nodule are in the range of the stator material thickness.
Fig. 3.10. Flux plot in the vicinity of the modified (nodules) stator slot area.
In Fig. 3.11 the FEM calculated magnetic flux density distributions on the surface of the rotor
under one slot pitch are illustrated both with a conventional stator slot opening and a modified
one. The modified stator slot opening, in accordance with Fig. 3.11, reduces the permeance
harmonics by half, compared to the same slot opening without the nodules.
81
0.88
0.9
0.92
0.94
0.96
0.98
1
1.02
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x / τ u
B / B max
1
2
Fig. 3.11. Flux density variation on the surface of the non-conducting rotor under one slot pitch, when no
current runs in the slot. 1) A normal slot opening, 2) the modified slot opening with nodules.
The ratio o/δ is 1.
The magnetically most effective way to eliminate the permeance harmonics seems to be the use
of a semi-magnetic wedge as a slot lock. Fig. 3.12 a) shows the mesh plot of the slot opening
with a modified slot wedge and in Fig. 3.12 b) the flux lines can be seen. The semi-magnetic
wedge is a material with a low relative permeability, which is usually between two and five. As
the figure illustrates, the wedge has to thrust out of the slot opening into the air-gap to ensure
the best possible result. When a low permeability wedge is formed like a magnetic lens, it
guides the flux lines to produce a uniform flux density on the rotor surface. Also this method
has some drawbacks. The manufacturing of a long stick with a rather accurate form is quite
difficult. Also the durability of the wedge material may, in some cases, be doubted.
Fig. 3.13 describes the flux density variation under a slot pitch on the surface of the rotor, when
different kinds of stator slot opening modification methods are used. Traditional slot wedges
with different permeabilities, lens formed wedges with two low permeabilities as well as
nodules have been studied. Both of the solutions - a high permeability in the slot wedge (µr =
10) or a low permeability (µr = 2) and modified wedge geometry - give a good result. It is
82
possible to eliminate the permeance harmonics almost totally. The nodules give a satisfactory
result by reducing the flux drop to half of the original one.
a)
b) Fig. 3.12. a) Mesh plot in the vicinity of a stator slot opening with the special slot wedge introduced in
this work. b) Magnetic flux lines under one stator slot pitch when the new slot wedge is used.
0.88
0.9
0.92
0.94
0.96
0.98
1
1.02
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x / τ u
B / B max
1
2
3
4
5 67
Fig. 3.13. Flux density variation on the surface of the non-conducting rotor under one slot pitch with
different variations of the slot opening, when no current runs in the slot. The ratio o/δ is 1. 1) a
normal slot opening, 2) a traditionally formed, straight slot wedge µr = 2, 3) a slot wedge µr =
5, 4) a slot wedge µr = 10, 5) a new, lens-type slot wedge µr = 2 with 0.75 mm arc into the air-
gap, 6) a new lens-type slot wedge µr = 5 with 0.2 mm arc into the air-gap, 7) a slot opening
with nodules.
83
Let us examine the behaviour of the semi-magnetic wedge when the slot to be analyzed carries
the peak phase current. Fig. 3.14 a) illustrates the magnetic flux lines in the stator slot opening
region when the slot opening has no wedge, and Fig. 3.14 b) illustrates the magnetic flux lines
in a stator slot opening region when the slot opening has a semi-magnetic wedge, µr=5. In Fig.
3.15 the flux density variation on the rotor surface is shown without a wedge and with wedges
having the permeabilities µr=2 and µr=5.
a)
b) Fig. 3.14. Magnetic flux lines in a stator slot opening region when the slot to be analyzed carries a
current, a) the slot opening has no wedge, b) the slot opening has traditionally formed straight
semi-magnetic wedge, µr=5.
As it was mentioned earlier, the winding arrangement and the form of the stator slot opening of
a solid-rotor induction motor have a great influence on the harmonic eddy currents on the
surface of the rotor. In table 3.2 the losses due to the harmonic eddy currents in the test machine
are calculated with the transient FEM analysis and with the analytical equations given above.
The calculated stator configurations are a full-pitch winding with a basic stator slot opening and
a short-pitch winding with a basic stator slot opening. A short-pitch winding was also calculated
with a slot opening with nodules and with different wedges. For the analytical analyses, the
factor to estimate the effects of the nodules and different wedges was determined from the
figures depicted in Fig. 3. 13. The factor is the ratio of the flux density sags. For instance, for
the traditionally formed straight wedge the factor is 0.03/0.11=0.273 (0.03 T is the flux density
sag in curve 3 and 0.11 T is the density sag in curve 1). This factor is used for defining the
magnetizing current of the permeance harmonic using Eq. (3.49). The fundamental rotor loss
due the slip is 4100 W. The per-unit slip is 1.5 %. The time step in the transient analysis was 10
µs.
84
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x / τ u
B / B max wedge, µ r = 2
wedge, µ r = 5
no wedge
Fig. 3.15. Flux density variation on the surface of the rotor under one slot pitch with different slot
wedges, when the motor peak current runs in the slot. The ratio o/δ is 1. The flux density
derivatives with respect to the tangential length are much smaller when the wedges are used.
Table 3.2. Calculated harmonic eddy-current losses on the surface of the solid steel rotor with different
stator designs for the test motor.
Full-pitch winding
Short-pitch winding, w/τp = 5/6
Short-pitch winding with a straight wedge (µr = 5)
Short-pitch winding with a modified wedge (µr = 5)
Short-pitch winding with nodules
FEM 2880 W 1820 W 1220 W 1110 W 1610 W Analytical 2790 W 1710 W 1150 W 1030 W 1360 W
The reduction of the losses, obtained by means of minimising the permeance harmonic, remains
quite low in this case. This may be due to the narrow slot openings and also due to the long air-
gap. The reference values refer to line 1 in Fig. 3.9, which already gives low losses in itself.
Figs. 3.16a and 3.16b describe the eddy current distribution at one moment of time on a surface
of the rotor with a full-pitch stator winding and with a 5/6-short-pitch winding, respectively.
While the fundamental current density is around 1.5 A/mm2, the harmonic eddy current density
arises over 20 A/mm2 with a full-pitch stator winding. With a 5/6-short-pitch winding the
maximum eddy current density stays at about 10 A/mm2.
85
a)
20 − 24 16 − 20 13 – 16 11 − 13 8 − 10 6 − 8 5 − 6
4.5 − 5 4 − 4.5 3.6 − 4
3.2 − 3.6 2.8 − 3.2 2.4 − 2.8 2.2 – 2.4 2.0 − 2.2 1.9 – 2.0
b)
12 – 14 10 − 12 9 − 10 8 − 9 7 − 8 6 − 7 5 − 6
4.5 − 5 4 − 4.5 3.6 − 4
3.2 − 3.6 2.8 − 3.2 2.4 − 2.8 2.2 − 2.4 2.0 − 2.2 1.9 – 2.0
Fig. 3.16. Eddy-current density on a surface of the rotor a) with a full-pitch stator winding, b) with a 5/6-
short-pitch winding. The numeral values are given in A/mm2. Please notice the different colour
scales in the figures. The calculations are performed with the time-stepping version of Flux
2D.
As a conclusion it could be stated that in solid-rotor machines the minimisation of the harmonic
losses is of vital importance. In a well-designed machine the harmonic losses may be reduced to
a level where the efficiency of the machine will not be diminished compared to commercial
squirrel cage machines of the same power. In the opposite case, the harmonic losses ruin the
performance of the machine.
86
3.1.4 Frequency converter induced rotor surface losses
As it was discussed earlier in the thesis, the solid-rotor surface is very sensitive to the air-gap
harmonic contents. The time and spatial harmonics of the air-gap flux density should be
minimised in order to avoid excessive rotor surface losses. The motor design has a great
influence on the harmonic content of the flux density, but the motor supply current should also
be as sinusoidal as possible. Present-day voltage source frequency converters, however, use
pulse width modulation (PWM) technology to produce the motor supply voltage. Lähteenmäki
(2002) used the square-wave frequency converter output and showed that in direct PWM
inverter supply the frequency modulation index (fsw/fs) must be at least 21 to achieve that the
motor losses remain lower than in the square-wave supply. Thus, in a 200 Hz machine drive the
switching frequency should be at least 4200 Hz.
Because the number of turns per phase in the stator windings of low-voltage, high-speed
machines is very low – typically, the number of turns in a coil can be one in such machines –
the motor transient inductance is also much lower than in conventional 50 Hz machines. A large
air-gap length and the rotor surface saturation in high-speed solid-rotor machines decrease the
transient inductance even more. As a consequence, the stator current follows rapidly the voltage
level changes in PWM-modulated supply voltage and the ripple of the stator current may
remain high even though the frequency modulation index is high. (Huppunen 1997)
Since present-day frequency converters in the power range relevant here can typically be used
at switching frequencies in the range of 1.5 kHz - 4 kHz without decreasing the converter rated
output power, it is useful to filter the output voltage of the inverters with moderate size
inductors and capacitors. The LC-filter inductor inductance value is typically selected to be half
of the stator leakage inductance. The capacitor is selected to compensate 80 % of the no-load
current. These values typically produce suitable resonance frequencies for the filter. In this
thesis, the motor supply voltages were filtered with a suitable sine-filter, which produces almost
purely sinusoidal voltages to the motor. The design and the advantages of the filter are
discussed more closely in Huppunen (1997) and Huppunen (2000a). Some measurement results
revealing the effects of switching frequency are given in Appendix B (Huppunen 1998a).
87
In the results reported in App. B. the fundamental rotor loss is approximately 5 kW at 250 kW
shaft power. When a 3.6 kHz switching frequency direct inverter supply and an inverter supply
with an LC filter are compared, the difference in motor no-load losses is approximately 6 kW.
The losses of the LC-filter itself are about 2 kW. Because the motor additional loss occurs
mainly in the surface of the rotor and the calculated spatial harmonic losses in a rotor surface
are according to table 3.2. about 1 kW, the inverter supply can double the rotor losses in high-
speed solid-rotor machines. This has a significant effect on the rating of the motor.
3.2 Friction losses
In high-speed machines the gas friction may also be of significant importance. Next, a brief
introduction to the friction losses will be given. The rotating rotor gives a tangential velocity
component for the air-gap gas. In addition, the gas has an axial velocity component if the
cooling gas is blown through the air-gap. Both the tangential and axial velocities affect the
friction torque of the rotor. Due to the high angular velocity, the estimation of the friction losses
is very important in the case of a high-speed machine. The friction losses in the air-gap can be
estimated by the equations for rotating cylinders in free space or in enclosures. E.g. Saari (1998)
has reported a quite comprehensive analysis on the friction of high-speed machines. Part of the
principles introduced in Saari’s study is repeated in the following.
The friction power Pfr associated with the resisting drag torque of a rotating cylinder is
lrCkP T43
1fr π Ωρ= , (3.55)
where CT is the torque coefficient, ρ is the mass density of the fluid, Ω is the angular velocity, r
is the radius, l is the length of the cylinder and k1 is the roughness coefficient (1.0 for smooth
surfaces and typically 2…4 for axially slotted surfaces). Because of the very complicated nature
of the gas flow in a slotted rotor surface the torque coefficient must usually be determined by
measurements.
When a cylinder is rotating in free space i.e. without the stator, one way to determine the nature
of the tangential gas flow exerted by the rotating cylinder is to use the tip Reynolds number that
determines the ratio between the inertia and viscous forces
88
µ
Ωρ 2
rrRe = (3.56)
where µ is the dynamic viscosity of the fluid.
In order to take the effect of the enclosure into account, the radial air-gap length has to be
included in the Reynolds number. This is done in the Couette Reynolds number, which is
µ
δρδ
uRe = , (3.57)
where δ is the radial air-gap length and u is the peripheral speed of the rotor.
The torque coefficient equations within the different flow regimes are (Bilgen 1973)
)10500(515.0 45.0
3.0
<<
⋅= δδ
δ
ReRe
rCT , (3.58)
)10(0325.0 42.0
3.0
δδ
δ
ReRe
rCT <
⋅= . (3.59)
In high-speed machines the tip Reynolds number is typically above 104, which means that these
machines are operated within a turbulent flow area.
Equations (3.58) and (3.59) have been tested with cylinders having relative air-gap lengths from
0.07 to 1, and the experimental data was within ±9% the calculated curve. When the relative
radial air-gap length increases, at some point the tangential flow is not affected by the stationary
outer cylinder any more, and the equations for free cylinders have to be used. (Saari 1998)
It should be remembered that the torque coefficients given are only valid for smooth cylinders.
Complete research work concerning the influence of rough air-gap surfaces on the friction
losses has not been published yet. Some estimation, however, can be made. Larjola (1991)
measured the friction losses in the air-gap of a high-speed generator. Both air-gap surfaces had
axial grooves. The author obtained a roughness coefficient of about 2.5. Larjola (1999) also
89
measured high-speed machines, which have smooth rotors and open stator slots or stator slots
closed with wedges. According to the research, a roughness coefficient for open stator slots is
only about 1.1. Thereby, the stator slots increase the friction losses in the air-gap only slightly
from those estimated for smooth air-gap surfaces, and the benefits obtained from the improved
heat transfer are surely higher.
The friction losses increase if there is an axial gas flow through the air-gap. The rotor forces the
cooling gas into a tangential movement and some power is needed for this acceleration.
For the axial gas flow through the air-gap the Reynolds number is
µ
δρ 2ma
vRe = (3.60)
where vm is the mean axial gas flow velocity in the air-gap.
When the radial air-gap length is small compared to the rotor radius, the power loss can be
approximated with
2m2afr, uqkP = , (3.61)
where k2 is the velocity factor and qm is the mass flow rate of the cooling gas. The mean
tangential velocity is usually expected to be half the rotor surface speed. According to the
studies of some authors, the theoretical velocity factor gets a value of 0.48. The real value is
anyway much lower. Larjola (1999) also studied the effect of the stator slots on the velocity
factor. According to his study, the velocity factor gets a value of 0.18 for a smooth stator
surface and 0.15 for a rough (slotted) stator surface. Thereby, the stator slotting decreases the
losses associated with the cooling gas flow through the air-gap. If the rotor is rough, the factor
can be expected to be close to the theoretical value.
The ends of the rotor do also have friction losses. The nature of the tangential flow is
determined with the tip Reynolds number. The power needed to rotate an end is
)(21 5
15
23
Endsfr, rrCP T −= Ωρ , (3.62)
90
where r2 and r1 are the outer and inner radii of the end, respectively. In electric machines the
free space for the rotor ends in the end-winding area is typically large, and the rotor end acts
like a centrifugal pump. When the rotor end is assumed to rotate in free space, the torque
coefficient is (Kreith 1968)
)103(87.3 5r5.0
r
⋅<= ReRe
CT , (3.63)
)103(146.0r
52.0
r
ReRe
CT <⋅= . (3.64)
3.3 Stator core losses
This thesis does not in detail concentrate on the stator losses. However, some general basic
knowledge on the stator iron losses is repeated here. The time varying fluxes produce losses in
ferromagnetic materials, known as core losses. It has been generally accepted for a long time
that the average iron power loss per unit volume pFe consists of a sum of a hysteresis power loss
phys and a dynamic (eddy current) power loss pdyn
dynhysFe ppp += .
The dynamic iron power loss can be divided into the classical eddy current loss pclass and the
excess or anomalous loss pexc
excclassdyn ppp += . (3.65)
The hysteresis loss results from the discontinuous character of the magnetization process at a
very microscopic scale and is equal to the area of the quasi-static hysteresis loop times
magnetizing frequency. The classical loss is associated with the macroscopic large-scale
behaviour of the magnetic domain structure. The excess loss is caused by the domain wall
motion, which generates the local eddy currents in the vicinity of the moving walls, and by the
wall interaction with lattice inhomogenities (Saitz 1997).
The flux density variation can be alternating or rotational. If it is alternating, it can be sinusoidal
or non-sinusoidal (distorted). From the hysteresis point of view, this non-sinusoidal variation
91
can be such, that it causes minor loops to the BH-characteristics of the material. Rotational
distribution of the flux density can be classified as purely rotational or elliptical. In the case of
the alternating flux there is a quite stable theory, which is based on Epstein loss data of the
material. As for the rotational loss, the situation is not so clear and the problem of measuring
and calculating the rotational loss has not yet been completely resolved (Saitz 1997).
In rotating machines, the flux patterns in the core may vary in a complicated way. Iron losses in
rotating machines occur due to the alternating, high frequency and rotating fluxes. The
alternating flux occurs predominately along the outer periphery of a stator yoke and in the stator
teeth. High frequency fluxes occur in the stator teeth. Circular flux polarization occurs at the
roots of the stator teeth and elliptical at the back of the stator slots. As a consequence, it has
been estimated that over 50 % of the iron losses in an induction machine are caused by the
rotating magnetic flux conditions. A rotational flux in the plane of the machine laminations
causes iron losses, which far exceed those caused by the alternating flux (Findlay 1994). It can
be observed that, in order to accurately solve iron losses in a rotating machine, a very
complicated model would be needed. Thus, a lack of accuracy exists and this can be attributed
to: (Bertotti 1991)
1. Rough estimation of the flux density distribution and the flux polarization.
2. Differences in geometry and supply conditions with respect to the standard Epstein
test.
3. Harmonics in the flux due to the iron saturation and teeth frequencies.
4. Modification of the magnetic properties of the material due to the residual and applied
stresses associated with the lamination punching and core assemblage.
Iron losses have been found to be proportional to the time derivative of the flux density
.dd,
dd,
dd 2
3
exc
2
classhys
∝
∝∝
tBp
tBp
tBp (3.66)
The hysteresis loss can be approximately calculated using an empirical relationship from
Steinmetz that ∫ = nBCBH maxhd so that
92
nBfCp maxhhys = . (3.67)
The values Ch and n are determined by the nature of the core material. The exponent n may vary
between 1.5 and 2.5 for different materials and is actually a function of Bmax in a given core.
According to Faraday’s induction law, the alternating flux induces an electromotive force in the
core, which in turn produces eddy currents that circulate in the iron. These eddy currents oppose
the alteration of the flux. The iron in the magnetic circuits is laminated to prevent excessive
eddy currents. A piece of lamination with a thickness τ is considered. The classical model
assumes a magnetization process perfectly homogenous in space and a sinusoidal flux
waveform. In the range of magnetizing frequencies where the skin effect is negligible, i.e. the
penetration depth is much larger than the lamination thickness τ, the classical eddy current loss
can be expressed as (Bertotti 1988)
ρ
τ6
π 2max
222
classBfp = . (3.68)
The Epstein test data are more appropriately used for the design of transformers since in
transformers the flux polarization is alternating. However, in rotating machines, a large portion
of the machine core is magnetized under rotational flux conditions. This poses a dilemma to the
machine designer since he must use the Epstein test results to predict the core losses. The
machine designer usually tries to circumvent the problem by introducing empirical loss
correction factors that are defined through practical experience. Such an approach becomes
increasingly inadequate as soon as novel concepts for machine designing are introduced and
higher working frequencies are attained (Bertotti 1991).
From this brief survey of the iron losses in rotating field electric machines, it can be concluded
that it is not possible to derive simple and exact analytical expressions for the iron losses in a
rotating electric machine. Besides, it is not the aim of this thesis to make a detailed investigation
of the iron loss. Therefore, in this study, a simple and well-known expression for the specific
iron loss is used.
The fixed Steinmetz law is the result of long-term industrial experience in the field of rotating
machines and is written (Vogt 1996)
93
2maxhyshyshys Hz50
Bfkp σ= , (3.69)
where σhys is the hysteresis loss of the material per weight at 50 Hz frequency and 1 T flux
density, and khys is an empirical coefficient, which takes the distortion of the magnetic flux
density into account.
Empirical equations are derived also to the eddy current loss. Vogt (1996) gives an equation
2max
2
ececec Hz50Bfkp
= σ , (3.70)
where σec is the classical eddy current loss of the material per weight at 50 Hz frequency and 1
T flux density, and kec is an empirical coefficient, which takes the distortion of the magnetic
flux density into account.
The total fundamental core losses are the sum of the fundamental hysteresis- and classical eddy
current losses:
2
max2
ecechyshysechysFe T1Hz50Hz50
+=+=
Bfkfkppp σσ . (3.71)
In combining these we get extrapolation equations for the core loss:
Fe
2max
Fe0.1Fe T1Hz50mBfkpP
n
= or Fe
2max
Fe5.1Fe T5.1Hz50mBfkpP
n
= , (3.72)
where n is depending on the material (relation between hysteresis- and eddy current losses), and
it varies between 1.4 – 2. For example, for a high-frequency electric steel sheet M250-50A n =
1.609. The coefficients p1.0 and p1.5 are the core losses of the material per weight at 50 Hz
frequency and at 1.0 T or 1.5 T flux density respectively, and p1.0 = σhys + σec. Table 3.3 gives
an example of empirical core loss coefficients for induction motors.
94
Table 3.3. Example of empirical core loss coefficients for induction motors (Vogt 1996).
Tooth Yoke
kFe khys kec kFe
1.8 1.5 1.8 1.5 - 1.7
3.3.1 Stator lamination in high-speed machines
An analysis of equation (3.70) indicates that after the motor frequency has been fixed, the eddy
current loss may be prevented by choosing the high resistivity iron sheets as thin as possible.
The iron sheets used in the stator core of a 50 Hz AC-motor do not suit well for a high-speed
machine, since the materials of the latter have too large core losses. In high-speed machines the
eddy current loss become dominant, thus thin laminations are preferred. The sheet thickness is a
trade-off between iron losses and manufacturing costs. Table 3.4 compares 0.35 mm and 0.50
mm laminations at a 100 Hz, 200 Hz and 400 Hz frequency. The increase in the specific total
loss is 14 % at 100 Hz, 26 % at 200 Hz and 37 % at 400 Hz when the iron sheets are changed
from 0.35 mm to 0.50 mm laminations.
Table 3.4. Specific total loss of some stator sheets (Cogent power Ltd 2002).
M250-35A (0.35mm) M250-50A (0.50mm)
Specific total loss [W/kg] @ 100 Hz, 1 T 2.41 2.75 Specific total loss [W/kg] @ 200 Hz, 1 T 6.14 7.73 Specific total loss [W/kg] @ 400 Hz, 1 T 17.1 23.4
3.4 Resistive losses of the stator winding
Resistive losses of the stator winding are
2SSCu.S IRmP ⋅= , (3.73)
where RS is the resistance of the stator winding per one phase. The DC-resistance of the stator
phase winding can be determined by
j
mS(DC) S aA
lNRσ
= , (3.74)
95
where NS is the number of the turns of the stator winding per phase, lm is the length of one turn
of the winding, σ is conductivity of the winding material, a is the number of parallel conductors
and Aj is the cross-section of one conductor. The length of the coil depends on the stator core
length LS, pole pitch τp, chord factor χ and the average distance of the coil turn-end Eew (Vogt
1996):
)2(2 ewSm ELl p ⋅++⋅≈ χτ , (3.75)
where χ = w/τp. When the alternating current is flowing through a conductor, according to the
Amperes law, the magnetic field strength curl occurs around the current. This time varying
magnetic field crowds the current on the surface of the conductor. The phenomenon leads to an
unequal distribution of the current across the conductor cross-section, and this is known as the
current skin effect. The skin effect increases the conductor resistance and thus also the winding
losses.
Stator windings are usually made of several parallel conductors, so that there may be dozens of
parallel conductors in a single stator slot. The current in the nearby conductor causes a time
varying magnetic field and induces a circulating current inside the conductor. This
concentration of current due to the presence of neighbouring currents is called proximity effect.
This phenomenon also increases the stator winding resistance and thus also the winding losses.
The penetration depth is very closely related to the current pinch effect. The penetration depth
determines the distance, where the electromagnetic wave is alleviated to 1/e of the original
value. The penetration depth, also known as skin depth, dp is defined as
f
dσµπ1
p = . (3.76)
The analysis can be done for a round conductor based on the Bessel-function solution, but the
round conductor can be replaced without losing accuracy with a square-shaped conductor which
has an equal cross-sectional area, a width equal to 2/π cdt = , if the conductor diameter dc is
smaller than the skin depth. Then, the analysis can be in rectangular coordinates (Ferreira
1994).
96
The packing factor η is defined as follows (Ferreira 1994),
bt
t+
=η , (3.77)
where b is the distance between the bare conductors. The ratio between the AC- and DC-
resistance is according to Ferreira (1994)
+−
−+−+
='cos'cosh'sin'sinh
2')12(
coscoshsinsinh
222
DC
AC
ζζζζζη
ζζζζζ n
RR , (3.78)
where n is the number of turn layers in a slot and
p
c
2πdd
=ζ , and ηζζ =' ; (3.79)
In Eq. (3.78) the first term of the equation determines the effect of the current concentration and
the second term is the proximity effect. For example, in a 200 Hz machine that has 15 turn
layers in a winding and a turn is 1.0 mm, the AC-fields increase the resistance by 2 %,
according to Eq. (3.78).
3.5 Loss distribution and optimal flux density in a solid-rotor
high-speed machine
Several authors (e.g. Kim 2001) stated that the flux density levels should be decreased in high-
speed machines in order to obtain the highest possible efficiency. This may be the right
statement, if the frequency is high enough, e.g. more than 400 Hz, or the rotor has a squirrel
cage, which forges a high torque at a low slip. But, for a slitted solid steel rotor supplied at a
lower than 300 Hz frequency, a high air-gap flux density should be used so that the maximum
torque can be attained. The flux density values for the stator laminations are to be chosen so that
they are close to the values of conventional 50 Hz machines, and, therefore, high quality stator
sheets must be used. High stator flux density values give, as a result, also high air-gap flux
density values, which proved to be prerequisite in order to get a high performance solid-rotor
machine.
97
A theoretical examination was done with the test machine. The number of turns in series per
stator winding was varied in order to have different flux densities in the motor. The stator slot
geometry was also varied in order to get minimal stator losses. In Fig. 3.17 the loss distributions
and the output powers are drawn at the constant power loss.
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
11000
12000
0.717 0.762 0.813 0.871 0.938 1.016Air-gap flux density [T]
Pow
er lo
ss [W
]
264
267
270
273
276
279
282
285
288
291
294
297
300
Out
put p
ower
[kW
]
P R,harm
P R,Cu
P S,Cu
P S,Fe
P fr
P shaft
Fig. 3.17. Loss deviation and the output power of the test motor at the constant loss power when the air-
gap flux density is varied.
It can be noticed that, when the air-gap flux density increases from 0.7 T to 0.9 T, the motor
output power increases by about 30 kW, which is more than a 10 % increase in the output
power. The efficiency (bearing friction and ventilation losses excluded) of the machine
increases at the same time from 0.956 to 0.961. The beneficial effect as a consequence of
increasing the flux densities in the solid-rotor machine may be regarded as an interesting
discovery. Traditionally, as higher speed machines are concerned, low flux density values have
been suggested.
3.6 Recapitulation of this chapter
Because of the high power density in high-speed machines, the power loss prediction has a
consequential role in the designing of a high-speed machine. For high-speed machines, the
significance of the harmonic losses grows and, especially for solid-rotor machines, the
harmonic loss on the surface of the rotor is of substantial importance. The main goal in this
98
chapter was to introduce an analysis of the calculation of the harmonic losses in a solid rotor
due to a non-sinusoidal magnetomotive force in an air-gap of a machine. The analysis
introduced by Bergmann gives results, which are converging with the results given by the FEM.
Several means to diminish these harmonic losses are also investigated. The results show that the
chorded two-layer winding decreases the rotor harmonic losses effectively in solid-rotor
machines. By modifying the stator slot opening the permeance harmonics can be reduced.
The air-gap windage loss is considerable in high-speed machines. A digest of the calculation
analysis for the air-gap windage introduced by Saari (1998) is given. The stator iron and copper
losses are processed briefly by a traditional linear calculation method using equivalent circuit
parameters, the same as are used with conventional speed machines, but some high frequency
aspects are also included. At the end of the chapter, a typical loss distribution in a 150 Hz high-
speed solid-rotor machine is shown and a new concept to choose the flux densities for this
machine type is also given.
99
4. Electromagnetic design of a solid-rotor induction motor
It is usually believed that a solid-rotor induction motor has poor electrical properties but,
according to the author’s experience, good, even excellent, drive properties can be reached
when the machine is properly designed.
4.1 Main dimensions of a solid-rotor induction motor
In the following, the solid-rotor design is studied in order to find the best solid-rotor
construction under the given constraints. This study concentrates on the slitted solid-rotor
structure with copper end rings (rotor c in Fig. 1.2).
4.1.1 Utilization factor
The utilization factor is an important design coefficient that indicates the internal apparent
power Si of the rotor volume and the motor electrical frequency. The equation for the machine
internal power with utilization factor C, stator bore diameter Dδ, electric length of the machine
L’ and synchronous speed ns=f /p expresses
s2
1
2
02
i 'ˆ2π' nLDBAnLCDS δδξ== , (4.1)
where A is the stator linear current density, ξ1 is the fundamental winding factor and δB is the
peak value of the air-gap flux density (Vogt 1996).
According to equation (4.1) the utilization factor for an induction machine is
δξ BAC ˆ2π
1
2
= . (4.2)
100
The stator linear current density A is a fictitious current sheet, which lies on the surface of the
stator or the rotor. This current sheet conveys a definite amount of current per stator inner
surface tangential length unit. The stator linear current density is defined with the stator
fundamental current effective value IS as (Vogt 1996)
ppImN
DImN
QDNIIA
ττ δδ
SSSS
S
uS
Su,
u
π2
π ==== . (4.3)
The mechanical output power of the machine can be calculated from the air-gap power by
taking the (fundamental) power factor cosϕ1 and the efficiency η into account (Vogt 1996)
s2
mech1s2
1
2
1mech 'cos'ˆ2πcos nLDC
EUnLDBAP
EUP δδδδ ϕηξϕη === . (4.4)
Therefore, the mechanical utilization factor of the induction machine can be found as
s
2mech
1mech 'cos
nLDPC
EUC
δ
ϕη == . (4.5)
Several authors and manufactures give the curves of the utilization factor for standard
machines, e.g. Vogt (1996). Now, a corresponding figure is constructed for two-pole high-speed
solid-rotor induction motors in accordance with the motors, which have been tested at
Lappeenranta University of Technology (LUT), Fig.4.1. It is also estimated that the mechanical
utilization factors rise similarly as they do in a traditional normal speed machine. Because
different rotor constructions produce very much different powers and losses, an own curve for
every rotor construction type is needed.
The maximum allowed linear current density of the machine strongly depends on the cooling.
The use of effective cooling systems may increase the allowed stress values up to twice the
values of a closed surface cooled machine. When the machine size is increasing, also the
cooling surface is increasing, and thus the allowed loads may be increased. Thereby, the
utilization factor is a function of the machine size.
The desired solid-rotor construction type has to be known by the designer in a very early state
of the design, because different kinds of solid-rotor constructions give a very large range of
101
motor output power and other properties. Therefore, the designer’s knowledge of the different
rotor structure characteristics is helpful. In Fig. 4.1 the utilization factors of the two-pole test
motors tested at LUT are drawn for different rotor constructions. This data reveal that the size
reduction discussed in chapters 1 and 3 may not totally be utilized because of the low torque
production capacity of the solid rotors compared to the traditional squirrel-cage induction
rotors. All the high-speed motors in question have an open air-cooling system. According to
Vogt (1996), this cooling system should increase the utilization factor by 30 percent compared
to the closed motor structure. Some test motors are, however, designed using lower stress
values than those of the traditional closed machines, and therefore the above-mentioned
advantage should not be used. The way in which the rated output power will be defined strongly
affects the utilization factor, since it is the rated output power that is used when the utilization
factor is defined. The rated power is usually defined by the temperature rise, which is specified
by the temperature classification of the winding insulation. Therefore, the power of the cooling
system has a great effect on the rated power. Here, the rated power is defined so that the
maximum efficiency is taken out of the machine at 75 percent of the rated power, although the
temperature rise would have allowed a higher output power than it was defined in this case. In
proceeding so, the curves may be compared. The utilization factor curve of standard two-pole
machines is also described in the figure. It is shown that the solid-rotor machine cannot reach
the utilization factors of a laminated rotor machine. The figure also illustrates that a cage
winding in a solid rotor increases the power of the machine in a noticeable degree compared to
the other solid-rotor constructions. The coated solid rotor can reach the level of a slitted solid
rotor with copper end rings. When no extra rotor winding material is used, the conductivity of
the rotor steel has a great impact on the characteristics of the solid-rotor machines (Pyrhönen
1991a).
The utilization factor of the machine is greatly affected by the air-gap magnetic flux density
used, since this has an influence on the torque produced. This is very significant, especially for
high-speed machines in which the air-gap flux density is not always an independently chosen
value. Usually, there is only one turn in every coil of the stator winding in a high-speed
machine, thus the air-gap flux density cannot be fixed easily by chancing the amount of turns of
the stator winding. This leads to the situation, where the stator inner diameter and the length
have to be fixed together with the air-gap flux density in order to get the desired output
characteristics. As a consequence, the utilization factor may vary in a wide range, and this
cannot be always considered to be a characteristic of a well-designed machine.
102
0
50
100
150
200
250
300
350
1 10 100 1000P shaft [kW]
C [k
Ws/
m3 ]
1
2
34 5
6
A
B
C
D
7
8
910
11
Fig. 4.1. Mechanical utilization factors for two-pole high-speed solid-rotor induction motors. The
motors are through cooled and the values are valid for motors with supply frequencies between
50 and 400 Hz. The numbers are corresponding with the numbers of the motors tested at LUT.
A) Solid rotor with copper cage (motors 3 and 4). B) Slitted solid rotor with copper end rings
(motors 6, 7, 9 and 10). C) Slitted solid rotor (motors 5, 8 and 11). D) Smooth solid rotor
(motor 1). The dashed line is the utilization factor curve for standard 50 Hz totally closed two-
pole induction motors.
In addition to the linear current density used, also the air-gap flux density affects strongly the
machine constant. Both the linear current density and the air-gap flux density depend evidently
on the cooling and thus the utilization factor gives only the machine power capacity at some
cooling dependent stress values. In other words, by changing the stress values it is possible to
get good solutions for different rated powers from the same rotor volume.
The lower utilization factor of the solid-rotor machine, compared to the laminated rotor
machine, is mainly due to the low power factor and larger rotor slip. If the cage winding is not
used, the lower efficiency also decreases the utilization factor. The large air-gap length and the
large phase angle of the conducting and magnetically non-linear solid-rotor material cause the
low power factor of the solid-rotor machines. When a laminated squirrel-cage rotor is used, the
power factor can reach 0.9, but with a solid-steel rotor together with copper end rings the power
factor remains at about 0.7. With a solid-steel rotor the power factor is about 0.65. When the
electrical utilization factor is concerned, the utilization factor is the same for the laminated rotor
103
machine and for the solid-rotor squirrel-cage induction machine. The power factor for machines
with other solid-rotor constructions remain below the upper mentioned alternatives, Fig. 4.2.
Fig. 4.2. Internal utilization factors for two-pole high-speed through-ventilated solid-rotor induction
motors. B) Solid rotor with copper squirrel cage. C) Slitted rotor with copper end rings. The
red line (A) is the utilization factor curve for standard 50 Hz totally closed two-pole induction
motors.
4.1.2 Selection of the L/D-ratio
The ratio of the stator core length L and the stator bore diameter Dδ has a strong influence on
the torque producing capability of the motor, in particular when a special solid-rotor structure is
used. The motor construction has a significant effect on the best rotor L/D-ratio. It must be
chosen optimally for every rotor type. If this ratio is badly chosen the motor does not achieve
the desired characteristics.
In the following, the test motor is calculated with different L/D-ratios in order to find the best
L/D-ratio. The motor has the following parameters, which are chosen to be constants, table 4.1.
Huppunen (1998b) investigated which are the best L/D ratios for a smaller solid-rotor machine.
The L/D ratio was calculated for a copper-cage solid rotor, for a slitted solid rotor with copper
end ring and for a slitted solid rotor without end rings.
100
150
200
250
300
350
1 10 100 1000Pshaft
[kW]
C [k
VA
s/m
3 ]
C
B
A
104
Table 4.1. Machine parameters.
Pole pairs 1 Stator voltage [V] 400
Nominal frequency [Hz] 150
Rotor volume [cm3] 8000 Number of stator slots 48 Number of rotor teeth 34
When the L/D-ratio is studied some assumptions have to be made. Here, the stator slots are
designed for every ratio so that the stator current density and the tooth flux density remain
constant. Also the stator yoke flux density is kept constant. In various rotors with different
amounts of slits the rotor slit depth is chosen so that the no-load peak flux density at the bottom
of the rotor teeth is in all cases the same. The rotor slit width is kept constant in all cases. For
reasons of mechanical strength the shaft underneath the end rings is 20 percent thicker than the
rotor core diameter underneath the slits. The axial length of the end ring is constant. The air-gap
flux density is kept constant which causes a theoretical and unreal study because the winding
parameters may adopt a form that is not manufacturable.
In this examination of the L/D-ratio will vary between 2.5 - 0.4, while the stator inner diameter
changes between 165 - 303 mm, increasing by 7 percent at every step, and the stator core length
varies from 411 mm to 122 mm. The main design parameters have the values given in Table
4.2.
Table 4.2. Designing parameters.
Stator current density, effective value [A/mm2] 4.5
Stator tooth flux density maximum [T] 1.55
Stator back flux density maximum [T] 1.45
Air-gap flux density maximum [T] 0.81
Rotor tooth no-load peak flux density [T] 2.0
4.1.3 Slitted rotor with copper end rings
In the following, the machine with a slitted steel rotor equipped with copper end rings is
examined. In Fig. 4.3 the shaft power is given as a function of the slip. The L/D-ratio affects the
power capacity because 300 kW can be taken out at a 0.9 % slip but, in the worst case, the slip
will be 1.9 %. The best L/D-ratio seems to be a very low value between 1.0 and 0.5. This is
105
understandable, because the conductivity of the copper in the ends of the rotor is very much
higher than the steel conductivity in the rotor bars.
0
50
100
150
200
250
300
350
400
450
500
0 0.5 1 1.5 2 2.5Slip [%]
Shaf
t pow
er [k
W]
0.400.490.600.740.901.111.361.662.042.49
L /D =
Fig. 4.3. Motor shaft power versus slip with different L/D-ratios when copper end rings are used in a
slitted solid-steel rotor.
Fig. 4.4 illustrates how the motor resistances change, while the L/D-ratio changes. It shows that
the rotor resistance decreases much more than the stator resistance increases and only with very
small L/D-ratio values their sum curve starts to increase. Meanwhile, the stator leakage
inductance increases from 36 µH to 80 µH. At least at small slip values, it seems that the motor
produces the best torque when the total resistance has the lowest value.
The machine efficiency curves in Fig. 4.5 can be studied as a function of the output power. It is
to be noticed that with this rotor type the L/D-ratio significantly affects the highest efficiency,
which varies between 95.0 and 95.6. This means a difference of 1.8 kW in the total loss power.
The best efficiency is achieved at a 280 - 320 kW output power. The efficiency is high when the
L/D-ratio is between 1.4 - 0.7, and the best L/D-ratio seems to be about 1.0.
106
0
2
4
6
8
10
12
2.49 2.04 1.66 1.36 1.11 0.90 0.74 0.60 0.49 0.40L / D
Res
ista
nce
[mΩ
]
0
15
30
45
60
75
90
Indu
ctan
ce [ µ
H]
R '
R
R
σS
R
S
tot
L
Fig. 4.4. Stator and rotor resistances when copper end rings are used in a slitted rotor.
94.9
95
95.1
95.2
95.3
95.4
95.5
95.6
95.7
200 250 300 350 400 450
Shaft power [kW]
Effic
ienc
y [%
]
0.400.490.600.740.901.111.361.662.042.49
L / D
Fig. 4.5. Motor efficiency curves when copper end rings are used in a slitted rotor.
Figs. 4.6 and 4.7 describe the motor losses and efficiency at a 300 kW output power. The stator
and rotor copper losses achieve their minimum when the L/D-ratio has its best value with
respect to the ability of generating the torque, but the rotor copper losses vary very rapidly and
they affect the optimal L/D-ratio while the stator copper losses change only a little. Also the
107
iron loss remains almost constant. The harmonic eddy current loss on the surface of the rotor
steel increases clearly when the rotor diameter increases. It is shown that the solid-rotor
machine equipped with copper end rings produces best when the L/D-ratio is small.
0
2000
4000
6000
8000
10000
12000
14000
2.49 2.04 1.66 1.36 1.11 0.90 0.74 0.60 0.49 0.40
Length / Diameter
Loss
pow
er [W
]
95
95.1
95.2
95.3
95.4
95.5
95.6
95.7
Effic
ienc
y [%
]
P R,harm
P S,Fe
P S,Cu
P R,Cu
P δ ,fr
η
Fig. 4.6. Motor efficiency and losses at an output power of 300 kW when copper end rings are used in a
slitted rotor.
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
5500
6000
6500
2.49 2.04 1.66 1.36 1.11 0.90 0.74 0.60 0.49 0.40
Length / Diameter
Loss
Pow
er [W
]
95
95.05
95.1
95.15
95.2
95.25
95.3
95.35
95.4
95.45
95.5
95.55
95.6
95.65
Effic
ienc
y [%
]
PcuSPCuRPfePharmRPfrPbearPcool.faneta
P S,Cu
P R,Cu
P S,Fe
P R,harm
P windage
η
P bearings
P cooling
Fig. 4.7. Motor efficiency and losses at an output power of 300 kW when copper end rings are used in a
slitted rotor.
108
4.1.4 Effects of the end-ring dimensions
The cross-section area of the copper end ring has a very significant effect on the rotor
characteristics. Fig. 4.8 illustrates the rotor slip, the fundamental rotor power loss, the current
density at the end ring and the efficiency versus the cross-section area of the copper end ring at
a 300 kW shaft power of the test motor.
0
1
2
3
4
5
6
7
8
9
10
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Cross-section area of the copper end ring [mm2]
Slip
[%],
Fund
amen
tal r
otor
loss
[kW
], C
urre
nt d
ensi
ty /
2 [A
/mm
2 ]
93.8
94
94.2
94.4
94.6
94.8
95
95.2
95.4
95.6
95.8
Effic
ienc
y [%
]
slip
fundamental rotor loss
efficiency
current density / 2
Fig. 4.8. Rotor slip and the fundamental rotor power loss, the end ring current density and the motor
efficiency as a function of the cross-section area of the copper end ring in a slitted solid rotor.
L/D = 1.4.
Fig. 4.9 gives the shaft power and efficiency as a function of the cross-section area of the
copper end ring at a 15 kW total power loss. Now, it is obvious that, if the cross-section area of
the copper end ring is larger than 600 mm2, the influence on the shaft power is negligible.
109
93.8
94
94.2
94.4
94.6
94.8
95
95.2
95.4
95.6
95.8
96
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Cross section area of Copper end ring [mm2]
Effic
ienc
y [%
]
210
220
230
240
250
260
270
280
290
300
310
320
Shaf
t pow
er [k
W]
Shaft powerEfficiency
Fig. 4.9. Motor efficiency and the output power as a function of the cross-section area of the copper end
ring in a slitted-solid rotor. L/D = 1.4.
4.2 Design of slit dimensions of a solid rotor
In the following, the slitted solid rotor is studied in order to find the correlation between the
motor performance and the rotor parameters with varying number of slits and slit dimensions.
To find the optimal slitted solid-rotor design, the analytical solving method cannot be used
directly, since the model with substitute parameters are differing in a considerable way from the
real electromagnetic phenomena of the machine. Thus, a FEM analysis is applied and the results
are compared to those calculated by using the analytical method. Axial rotor slits are used to
improve the performance of the solid-rotor induction motors. The rotor parameters analyzed in
the study are the number, the width and depth of the slits through which the best torque is
achieved. The other two parameters being of interest may be the efficiency and the power factor
of the machine, but these are left out of the study here, since it is obvious that the produced
torque and the slip frequency are of crucial importance when the efficiency and power factor
are defined.
The examination proceeds as follows. Several geometrical versions of the device are studied.
For each version interesting parameters or objective functions are solved and the best version is
selected. This chapter introduces the calculated FEM models and compares the calculation
results.
110
4.2.1 Solving the magnetic fields of a solid-rotor induction motor by
means of the FEM-analysis
The FEM analysis offers the possibility to solve the Maxwell equations of the magnetic field
problem numerically in a complicated geometry. But, the solution of the complete, three-
dimensional magnetic field of an induction motor is still too demanding a problem for present-
day computers. Some simplifications have to be made in order to keep the calculation time at an
acceptable level.
The magnetic field in the core of the machine is assumed to be two-dimensional. The three-
dimensional end-region fields of the stator are modelled approximately by using constant end-
winding impedances in the circuit equations of the windings. The laminated stator iron core is
modelled as a non-conducting, magnetically non-linear medium. The rotor steel is treated as a
conducting, magnetically non-linear material. The hysteresis is neglected at both media and is
taken into account in the stator only in the post-processing. The rotor hysteresis is neglected.
The highly conducting end rings are presumed, and the effects of the end rings are taken into
account by decreasing the conductivity of the rotor by an end-effect factor, chapter 2.5.
The time variation of fields in an electric machine is practically never sinusoidal, thus the non-
linearity of steel and the rotation of the rotor require the use of the time-stepping method to
accurately solve the magnetic field. This is a very time consuming process. The relatively long
time constants associated with the windings of the induction machine complicate the use of the
time-stepping method in the simulation of the steady-state operation. If the zero field is taken as
the initial state, tens of periods of the nominal frequency have to be simulated before a steady
state is reached. The results of the sinusoidal approximations or DC field calculations can be
used to find an initial state that is nearer to the steady state. If the time dependence of the field
is assumed to be sinusoidal, the computation time can be reduced radically. For this reason, the
assumption of sinusoidal time variation is commonly used, especially when effective steady-
state values are calculated. (Arkkio 1987).
The main problem in the calculation of a rotating machine is the question how the motion of the
rotor should be modelled. The accurate way of solving this question would be to use the time-
stepping method, but, as mentioned above, this method is often too time consuming for routine
computation. The easiest method to take the rotor motion into account in the sinusoidal
approximation is to treat the rotor as a quasi- or pseudostationary object. In the
111
pseudostationary approximation the rotor is fixed and the motion is modelled by multiplying the
conductivities of the rotor by the per-unit slip s. (Arkkio 1987).
Arkkio (1987) calculated a squirrel-cage and a solid-rotor induction motor with the time-
harmonic method and with the time-stepping method. In addition, the motors were measured.
Irrespective of the fact that at large slip values there are large errors in the torques obtained with
the time-harmonic method through the pseudostationary approximation, the values are very
close to the values obtained with the time-stepping method and with the measurements when
the slip is smaller than 10%.
The Flux2D –software by Cedrat was used. This software includes both magnetodynamic i.e.
time-harmonic and transient i.e. time-stepping solvers. The time-harmonic solver was chosen,
but some of the results were checked with the transient calculation. The solid rotor in a time-
harmonic solver may be modelled in two alternative ways. The rotor can be modelled as a solid
conductor by adding it to the circuit model. This requires the modelling of the whole motor,
which is thus a more time consuming alternative. The other way is to use the pseudostationary
approximation. In this case, it is necessary only to model one pole, which is thus a much faster
solution. Both the methods were tested and it was notified that the results were very much the
same. The pseudostationary approximation method was applied in most of the calculations of
this thesis.
Flux 2D offers several methods for computing the magnetic torque exerted on a part of the
device. The torque exerted in a given direction is obtained by differentiating the magnetic
energy W of the system with respect to a virtual displacement θ of the object in this direction.
θ∂
∂=
WTem . (4.6)
The magnetic energy is
vBWV
d21 2
∫=µ
(4.7)
112
The virtual work method allows computing of the torque exerted on parts that keep their shape
and that are surrounded by air. Therefore, regarding the air-gap torque of a rotating machine, it
is a very useful method.
The computation of the magnetic torque exerted on a ferromagnetic region can also be obtained
by integrating the magnetic pressure exerted on the boundary between this region and the
neighboring regions. This method is based on the Maxwell’s stress tensor. The electromagnetic
torque is obtained as a surface integral
( )∫∫
−⋅×=⋅×=SS
Sd211d 2
em nBBnBrSσrTµµ
, (4.8)
where σ is Maxwell’s stress tensor, r is a vector representing the radius of the rotor, its direction
and length from the rotor center to the point of calculation, and n is the unit normal vector of
the integration surface S. Both of the methods mentioned above are available in Flux2D used in
the FEM-calculations in this thesis. The Maxwell’s stress tensor is often criticized because of its
inaccuracy in numerical calculations. The virtual work method is often regarded as a more
reliable method and it is thus applied in this work.
In the time-harmonic calculation (magnetodynamic) the state variable (vector potential) is a
complex quantity. It varies sinusoidally in time, similarly as the derivative quantities vary. In
practice, when non-linear materials are present, the field and the magnetic flux density do not
vary sinusoidally. Therefore, in order that these non-linear materials are taken into account,
some approximations are applied. FLUX2D computes, starting from the user defined BH-curve,
an equivalent curve allowing the conversation of energy point by point. When a voltage source
is used, the points on the equivalent curve are calculated while it is supposed that the flux
density varies sinusoidally as a function of time. When a current source is used, the points on
the equivalent curve are calculated supposing that the field varies sinusoidally, Fig. 4.10.
113
B
HH
H(ω t)
B (ω t)
B
ω t
ω t
B
H
H
H(ω t)
B (ω t)
B
ω t
ω t
a) b)
B
H
c)
original curveB sinusoidalH sinusoidal
Fig. 4.10. BH-curve calculations in different time-harmonic models. a) flux density varies sinusoidally
(sinusoidal voltage supply), b) magnetic field varies sinusoidally (sinusoidal current supply).
c) Equivalent BH-curves.
It is important to notice, as time-harmonic solutions are used, that in some cases the harmonic
torques of the rotor may be incorrectly calculated. In such cases the correct torque value may be
obtained by calculating the torque at several rotor positions. The results must then be averaged.
Especially in the case of 32 rotor slits the time-harmonic solution fails in this case. It is
interesting to notice that a 48 stator slot 32 rotor slot combination is not traditionally
recommended (Richter 1954) because of possibly occurring synchronous harmonic torques at
positive speeds.
114
Fig. 4.11 reveals that the time-harmonic calculation produces oscillating torque results as a
function of the rotor position if the stator-slot − rotor-slot combination according to Richter
(1954) produces adverse synchronous torques at stall or at positive speeds.
The time transient (time-stepping) solver was thus also used to evaluate some cases. The
slowness of calculation performing, however, does not encourage the designer to use the time-
stepping analysis on a large scale.
272
274
276
278
280
282
284
286
0 1 2 3 4 5 6 7
rotor angle [degrees]
torq
ue [N
m]
283032343638404244465260
Fig. 4.11. Time-harmonic calculation results for the air-gap torque as a function of the rotor mechanical
angle. According to Richter (1954) for two-pole machines the combinations QS/QR = 48/30,
48/36, 48/42, 48/48, 48/54, 48/60 are adverse at stall and QS/QR = 48/32, 48/38, 48/44, 48/50,
48/56, 48/62 are adverse at positive speeds. The combinations belonging to the above
mentioned may produce oscillating results in the time-harmonic calculation. QS/QR = 48/28,
48/34, 48/40, 48/46, 48/52, 48/58 are recommended at positive speeds.
Fig. 4.11 illustrates that the time-harmonic torque calculation result of the rotor with 34 slits
behaves very smoothly as a function of the rotor angle and that the torque ripple shows the
worst values when the number of rotor slits is 32. These two cases were selected for evaluation
with the time-stepping transient calculation. The average torque values at 1.5 % slips calculated
with the transient method and the time-harmonic method are shown in table 4.3. The time step
in the transient analysis was 10 µs. The torque values obtained with both methods converge
extremely well to each other. Therefore, it seems that the time-harmonic method can be applied
at least when small slip values are considered. With the time-harmonic method, the calculation
115
was performed within a time of 20 to 30 minutes and, with the time-stepping method, the
duration of the calculation was 2000 to 3000 minutes. With the MLTM method, the calculation
was done within 1 to 2 seconds. A portable PC with a 1.8 GHz processor and 768 MB memory
was used.
Table 4.3. Torques at 1.5 % slip.
Transient calculation average torque [Nm]
Time-harmonic calculation average torque [Nm]
Rotor with 32 slits 276.9 278.3 Rotor with 34 slits 278.0 279.2
4.2.2 FEM calculation results
At first, the test machine was modelled in both ways, with a smooth and a slitted solid-rotor
structure. Since the motor has two poles, only half of the motor is modelled. The meshes of the
smooth and the slitted solid-rotor motor constructions are illustrated in Fig. 4.12.
a) b) Fig. 4.12. Meshes of the test motor equipped with a smooth solid-rotor and with a slitted solid rotor.
Flux penetration into the conducting rotor material causes eddy currents, which, again, tend to
prevent the flux penetration. The penetration depth in ferromagnetic, conducting material is
low, thus the flux is concentrating near to the surface of the rotor. When the rotor is axially
slitted, the slits increase the reluctance on the tangential flux path and the flux has to penetrate
116
deeper on its way to the other magnetic pole. The flux lines and the flux density distribution at
1.5 % slip are shown in Fig.4.13 and Fig. 4.14.
a) b) Fig. 4.13. Flux lines of a) a smooth solid-rotor and b) a slitted solid-rotor induction motor at 1.5% slip.
a) b)
2.47 – 2.64 2.31 – 2.47 2.14 – 2.31 1.98 – 2.14 1.81 – 1.98 1.65 – 1.81 1.48 – 1.65 1.32 – 1.48 1.15 – 1.32 0.99 – 1.15 0.83 – 0.99 0.66 – 0.83 0.50 – 0.66 0.33 – 0.50 0.17 – 0.33 2e-5 – 0.17
Fig. 4.14. Flux density distribution of a) a smooth solid-rotor and b) a slitted solid-rotor induction motor
at 1.5% slip.
The axial slits in a solid rotor form a fair path for the eddy currents to flow from one rotor end
to the other in the “rotor bars” between the slits. The current passing through the rotor tooth
creates, according to Ampere’s law, a magnetic flux circulating around the current path. Thus,
in the slitted rotor when the stator and rotor magnetic fields conflate, the flux lines form a
magnetic curl around the rotor current path in the rotor teeth. When the torque rotates the rotor
117
counter-clockwise, deep in the teeth, the flux is forced on the lagging sides of the teeth, and on
the leading sides of the teeth the flux density is very low, Fig. 4.15.
Fig. 4.15. Flux lines of two differently slitted solid-rotor induction motors at 1.5% slip.
The current penetration into the rotor material depends on the flux penetration. Fig. 4.16
explains in an illustrated way the benefit of slitting the rotor. While in a smooth solid rotor the
current concentrates only on the very surface of the rotor, in a slitted rotor the current spreads
out quite equally into the slitted area and the flux penetrates much deeper into the slitted rotor,
Fig. 4.16.
a) b)
1.15 – 1.22 1.07 – 1.15 0.99 – 1.07 0.92 – 0.99 0.84 – 0.92 0.77 – 0.84 0.69 – 0.77 0.61 – 0.69 0.54 – 0.61 0.46 – 0.54 0.38 – 0.46 0.31 – 0.38 0.23 – 0.31 0.15 – 0.23 0.07 – 0.15 0.00 – 0.07
Fig. 4.16. Current density distributions [A/mm2] of a) a smooth solid rotor and b) a slitted solid-rotor
induction motor at 1.5% slip. The time-harmonic solution is used and thus the rotor surface
harmonic current densities are not present. Please compare the result with the result of Fig.
3.16.
118
The distribution of the permeability of the rotor material pursues the current density
distribution, Fig. 4.17a. But, in a slitted rotor, where in the tooth area the flux density is low, the
permeability keeps its high value, Fig. 4.17b.
a) b)
2400 – 5000 2000 – 2400 1750 – 2000 1500 – 1750 1250 – 1500 1000 – 1250 800 – 1000
650 – 800 500 – 650 400 – 500 300 – 400 200 – 300 150 – 200 100 – 150 50 – 100 1 – 50
Fig. 4.17. Relative permeability distributions of a) a smooth solid-rotor and b) a slitted solid-rotor
induction motor at 1.5% slip.
Since the current penetration in the slitted rotor is considerably better than that in the smooth
solid rotor, axial slitting of a solid rotor increases the output torque up as high as about twice
the torque of the smooth solid rotor, as it is shown in Fig.4.18.
0
50
100
150
200
250
300
350
400
450
500
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Slip [%]
Torq
ue [N
m]
Slitted solid rotor
Smooth solid rotor
Fig. 4.18. Output torque of a smooth and a slitted solid-rotor induction motor.
119
4.2.3 Study of the rotor slitting
In the following, the slitted solid rotor design is studied in order to find out if varying the main
rotor design parameters has a substantial influence on the motor characteristics. The parameters
analyzed are the number of the rotor slits, the depth and the width of the rotor slits. The initial
rotor design parameters are according to the parameters given in table 4.5. Only the rectangular
rotor slit shape is studied and the depth of every second rotor slit may be lower than that of the
others. The mechanical limitations were not taken into account here. However in practice, the
rotor slit width is selected based on manufacturing aspects.
Table 4.5. The initial values of the rotor design.
Number of slits
Depth of slits [mm]
Width of slits [mm]
Cross-section area of end rings [mm2]
34 40 2.5 600 In the first phase the number of the slits was varied between 28 and 52 and the other parameters
were kept at their initial values. The motor was calculated at rotor slips 1.0 %, 1.5 % and 2.0 %.
The results are given in Fig. 4.19. The output torque as a function of the number of rotor slits
seems to have the form of a downwards-opening parabola the maximum value of which is
achieved when the number of the rotor slits is between 32 and 42, sliding to a higher number
when the slip increases. Anyway, the difference on the torque is considerably small as the
number of slits varies.
186
188
190
192
194
196
198
200
26 28 30 32 34 36 38 40 42 44 46 52
Number of rotor slits
Torq
ue [N
m]
a)
120
268
270
272
274
276
278
280
282
26 28 30 32 34 36 38 40 42 44 46 52
Number of rotor slits
Torq
ue [N
m]
b)
336
338
340
342
344
346
348
350
352
354
356
26 28 30 32 34 36 38 40 42 44 46 52
Number of rotor slits
Torq
ue [N
m]
c)
Fig. 4.19. Air-gap torque of the slitted solid-rotor induction motor as a function of the number of the
rotor slits. The rotor slips are a) 1.0 %, b) 1.5 % and c) 2.0 %.
In the following, both the number of the rotor slits and the width of the rotor slits are varied, but
the depths of the rotor slits are kept constant. As it is shown in Fig. 4.20, the combination of the
number of slits and the width of slits has a significant influence on the generated air-gap torque.
Surprisingly, it may be noticed that there exists an optimal number of rotor slits for every
individual width value of the rotor slits. Furthermore, when the optimal combination is chosen,
the product of the number of the rotor slits and the width of the rotor slits remains
approximately the same. Besides this, it seems that the rotor works best when the width of the
rotor slits is very narrow, i.e. 1 mm and the number of rotor slits is very high, but the width of
the rotor slits does not significantly influence the generated torque as long as the number of the
121
rotor slits remains within the given range. For example, if the width of the rotor slits is doubled
from 1 mm to 2 mm and if the number of the rotor slits is halved, the generated air-gap torque
will be almost the same. However, the manufacture of the slitted solid rotor is cost-effective
when the combination of the width and the number of the rotor slits are well selected. It should
also be remembered that the rotor frequency has a significant influence on the optimal rotor
slitting. As the rotor slip frequency fsl is low only a very light slitting is necessary, but the
higher the rotor slip frequency is the more rotor slitting is needed in order to achieve the best
capacity to generate the torque.
92
94
96
98
100
102
104
106
108
110
1.0 1.5 2.0 2.5 3.0Width of the rotor slits
Torq
ue
2832364046526068768492100112
a)
180
185
190
195
200
205
1.0 1.5 2.0 2.5 3.0Width of the rotor slits
Torq
ue
2832364046526068768492100112
b)
122
255
260
265
270
275
280
285
290
1.0 1.5 2.0 2.5 3.0Width of the rotor slits
Torq
ue
2832364046526068768492100112
c)
310
315
320
325
330
335
340
345
350
355
360
365
1.0 1.5 2.0 2.5 3.0Width of the rotor slits
Torq
ue
2832364046526068768492100112
d)
123
360
370
380
390
400
410
420
430
1.0 1.5 2.0 2.5 3.0Width of the rotor slits
Torq
ue
2832364046526068768492100112
e)
Fig. 4.20. Air-gap torque of the slitted solid-rotor induction motor as a function of the number of the
rotor slits and the width of the rotor slits. The rotor slips are a) 0.5 %, b) 1.0 %, c) 1.5 %, d)
2.0 % and e) 2.5 %.
In the next phase the influence of the depth of the rotor slits on the generated torque is
examined. Considering the manufacturing restrictions, the width of the rotor slits was set to 2.5
mm. According to the results got from the analysis of the rotor design parameters and given in
Fig. 4.21, the effect of the depth of the rotor slits on the generated torque is the most significant.
When all the rotor slits are kept in the same depth, the optimal depth is 50 mm, which is about
50 % of the rotor radius. If the depth of the rotor slits is left to 30 mm, the generated air-gap
torque is 15 % lower than in the previous case. The depth of the rotor slitting is restricted by the
saturation of the rotor material between the slits. In addition, the mechanical strength of the
rotor material limits the depth of the rotor slitting. To ease both these stresses the possibility
was analyzed to get deeper slitting by leaving every second slit lower than the others. However,
in doing so, only a 2 % improvement could be achieved.
124
80
85
90
95
100
105
110
30 40 50 60 40/20 50/30 60/30 60/40Depth of rotor slits [mm]
Torq
ue [N
m]
28323640
a)
160
170
180
190
200
210
30 40 50 60 40/20 50/30 60/30 60/40Depth of rotor slits [mm]
Torq
ue [N
m]
28323640
b)
125
240
250
260
270
280
290
300
30 40 50 60 40/20 50/30 60/30 60/40Depth of rotor slits [mm]
Torq
ue [N
m]
28323640
c)
310
320
330
340
350
360
370
30 40 50 60 40/20 50/30 60/30 60/40Depth of rotor slits [mm]
Torq
ue [N
m]
28323640
d)
126
360
370
380
390
400
410
420
430
30 40 50 60 40/20 50/30 60/30 60/40Depth of rotor slits [mm]
Torq
ue [N
m]
28323640
e)
Fig. 4.21. Air-gap torque of the slitted solid-rotor induction motor as a function of the number of the
rotor slits and the depth of the rotor slits. The rotor slips are a) 0.5 %, b) 1.0 %, c) 1.5 %, d) 2.0
% and e) 2.5 %.
Finally, it should be studied whether the number of stator slots does affect in any way the
optimal number of rotor slits. The test motor is calculated with 60 stator slots and the rotor slit
parameters are the original basic ones, 40 mm depth and 2.5 mm width. The number of rotor
slits is varying from 28 to 52. The results in Fig. 4.22 are analyzed. The optimal number of the
rotor slits remains between 34 and 42 regardless of the stator slot number.
190
191
192
193
194
195
196
197
198
32 34 36 38 40 42 44 46 48 50Number of rotor slits
Torq
ue [N
m]
a)
127
274
275
276
277
278
279
280
281
282
32 34 36 38 40 42 44 46 48 50Number of rotor slits
Torq
ue [N
m]
b)
348
349
350
351
352
353
354
355
356
32 34 36 38 40 42 44 46 48 50Number of rotor slits
Torq
ue [N
m]
c) Fig. 4.22. Air-gap torque of the slitted solid-rotor induction motor as a function of the number of the
rotor slits. The number of stator slots is 60. The rotor slips are a) 1.0 %, b) 1.5 % and c) 2.0 %.
4.2.4 Comparison of the FEM with the MLTM method
The calculation time needed by the analytical multi-layer transfer-matrix method is very short.
Each point in the torque-slip curve is calculated in one to two seconds. When the FEM is used
with a modern laptop PC the calculation time for each performance point is ranging from 15 to
30 min. For this reason, analytical calculation methods are really comfortable to use in every-
day designing work.
128
The torque characteristics of the test motor are shown in Fig. 4.23. The torque curves are
calculated by using the FEM, the MLTM with the curvature coefficient discussed in chapter 2.7
and the MLTM without the curvature coefficient. When the MLTM method is used in the
Cartesian coordinates the curvature coefficient must be used, otherwise the error in the
calculated torque varies between 20 – 50 % in the given slip range.
100
150
200
250
300
350
400
450
500
0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5
Slip [%]
Torq
ue [N
m]
FEM
MLTM with curvature co-efficientMLTM without curvature co-efficient
Fig. 4.23. Air-gap torque of the slitted solid-rotor induction motor as a function of the rotor slip
calculated with FEM and MLTM with or without the curvature coefficient.
In the following, the MLTM and the FEM results are compared when the number and the depth
of the slits are kept constant, 32 and 40 mm respectively, and the width of the slits is the variant.
The results are calculated with the rotor slip values between 0.5 % and 2.5 %, and are shown in
Fig. 4.24. Comparing the results obtained to those given by FEM, the MLTM method gives
higher values at very narrow slits and lower values at very wide slits. However, the maximum
error is less than 1.5 %. As mentioned earlier, the rotor frequency has an effect on the optimal
width of the slits, which is shown well also with the MLTM method.
129
101
102
103
104
105
106
107
108
109
1.0 1.5 2.0 2.5 3.0Width of the rotor slits [mm]
Torq
ue [N
m]
FEMMLTM
a)
191
193
195
197
199
201
203
1.0 1.5 2.0 2.5 3.0Width of the rotor slits [mm]
Torq
ue [N
m]
FEMMLTM
b)
130
250
255
260
265
270
275
280
285
290
1.0 1.5 2.0 2.5 3.0Width of the rotor slits [mm]
Torq
ue [N
m]
FEMMLTM
c)
300
310
320
330
340
350
360
1.0 1.5 2.0 2.5 3.0Width of the rotor slits [mm]
Torq
ue [N
m]
FEMMLTM
d)
131
340
350
360
370
380
390
400
410
420
430
1.0 1.5 2.0 2.5 3.0Width of the rotor slits [mm]
Torq
ue [N
m]
FEMMLTM
e) Fig. 4.24. Air-gap torque of the slitted solid-rotor induction motor as a function of the width of the rotor
slits. The number of rotor slits is 32 and the depth is 40 mm. The rotor slips are a) 0.5 %, b)
1.0 %, c) 1.5 %, d) 2.0 % and e) 2.5 %.
The varying parameter is changed according to the depth of the rotor slits and again a
comparison between the results given by the FEM and the MLTM method are made. It can be
noticed from the results shown in Fig. 4.25 that the MLTM method gives very good results
when compared to the results obtained by the FEM. The results differ only when the slits are
extremely deep.
90
95
100
105
110
115
30 40 50 60 40/20 50/30 60/30 60/40Depth of the rotor slits [mm]
Torq
ue [N
m]
FEMMLTM
a)
132
175
180
185
190
195
200
205
210
30 40 50 60 40/20 50/30 60/30 60/40Depth of the rotor slits [mm]
Torq
ue [N
m]
FEMMLTM
b)
250
255
260
265
270
275
280
285
290
295
30 40 50 60 40/20 50/30 60/30 60/40Depth of the rotor slits [mm]
Torq
ue [N
m]
FEMMLTM
c)
133
320
330
340
350
360
370
380
30 40 50 60 40/20 50/30 60/30 60/40Depth of the rotor slits [mm]
Torq
ue [N
m]
FEMMLTM
d)
380
390
400
410
420
430
440
30 40 50 60 40/20 50/30 60/30 60/40Depth of the rotor slits [mm]
Torq
ue [N
m]
FEMMLTM
e)
Fig. 4.25. Air-gap torque of the slitted solid-rotor induction motor as a function of the depth of the rotor
slits. The number of rotor slits is 32 and the width of the slits is 2.5 mm. The rotor slip is a) 0.5
%, b) 1.0 %, c) 1.5 %, d) 2.0 % and e) 2.5 %.
In the next phase, the number of the rotor slits is the variant, while the depth and the width of
the rotor slits are constants, 40 mm and 2.5 mm, respectively. The results in Fig. 4.26 show that
the MLTM method does not notice the optimal number of the rotor slits. It may also be
concluded that the MLTM calculates the torque to be accurately near to the optimal number of
rotor slits but also that, especially when the slip increases, the MLTM calculates too much
torque at a low number of rotor slits and too little torque at a high number of rotor slits.
134
184
186
188
190
192
194
196
198
200
202
26 28 30 32 34 36 38 40 42 44 46 52Number of rotor slits
Torq
ue [N
m]
FEM
MLTM
a)
264
266
268
270
272
274
276
278
280
282
284
26 28 30 32 34 36 38 40 42 44 46 52Number of rotor slits
Torq
ue [N
m]
FEMMLTM
b)
135
336
338
340
342
344
346
348
350
352
354
356
26 28 30 32 34 36 38 40 42 44 46 52Number of rotor slits
Torq
ue [N
m]
FEMMLTM
c)
Fig. 4.26. Comparison between the FEM and the MLTM results of the calculated air-gap torque of the
slitted solid-rotor induction motor as a function of the number of the rotor slits. The rotor slips
are a) 1.0 %, b) 1.5 % and c) 2.0 %.
4.3 Measured results
The measurement arrangements are documented in Appendix B. The test motor with the L/D-
ratio of 1.4 was tested with two rotors. The first rotor has 34 slits of which 17 pieces are 40 mm
and 17 pieces 20 mm deep, all of a 2.5 mm width. The second rotor has 17 slits of 50 mm depth
and 17 slits of 30 mm depth, all 2.5 mm wide. Fig. 4.27 shows a measured torque versus the
slip of the rotors. The calculated curves are also drawn in the figure. When the slip curves are
measured, it is impossible to keep the rotor temperature accurately constant, since the load
varies from no-load to rated load. The stator temperature was about 20 K lower in no-load point
than in the maximum torque measured point. In rotor steel material the temperature coefficient
of the resistivity is high, thus the temperature has a noticeable effect on the results. Hence, the
rotor with 50 / 30 mm deep slits was also calculated at a 30 °C temperature, and one point was
measured approximately at the same temperature. It is obvious that the rotor with 50 / 30 mm
deep slits has a better capacity of generating the torque, which is also confirmed by the stator
temperature. The stator temperature measured in the 50 / 30 mm slitted rotor was about 10 K
lower than the stator temperature measured in the 40 / 20 mm slitted rotor.
136
0
50
100
150
200
250
300
350
400
450
0 0,5 1 1,5 2 2,5
Rotor slip [%]
Torq
ue [N
m]
Slits 50/30, measured
Slits 50/30, calculated
Slits 40/20, measured
Slits 40/20, calculated
Slits 50/30, measured cold
Slits 50/30, calculated cold
Fig. 4.27. Measured and calculated torque versus slip curves of the 50 / 30 mm slitted and 40 / 20 mm
slitted rotors.
4.4 Discussion of the results
The results achieved with the multi-layer transfer-matrix method coincide satisfyingly with the
results got from the FEM. Only the extreme conditions cause results, which are not accurate
enough. Such conditions are e.g. a very low or high number of rotor slits or extremely narrow
slits or teeth.
The most significant factor that complicates the comparison between the measured and
calculated results is the temperature of the rotor. Since the rotor volume is low and the power
density is high in high-speed machines, temperature changes may happen very rapidly. Because
the temperature coefficient of the rotor material is very high (almost three times the coefficient
of copper), the meaning of the temperature is very important while measuring the rotor slip
frequency. In practise, it is not possible to measure the rotor speed at the same rotor temperature
from a low load to a heavy load. The difference of the slip between a cold and hot rotor may be
as high as 50 %. The temperature approximation of 150 °C used in the calculation for the rotor
seems to give acceptable results.
When well-conducting end rings are used in a solid rotor, the motor air-gap diameter per length
ratio should be much lower than that in a conventional squirrel cage induction motor. The
137
typical value for this ratio is 1.5 in a traditional two-pole machine. When in a solid rotor well-
conducting end rings are used the ratio should be approximately 1.0.
The difference in the produced torque is very low with a large number of rotor slits. The
number of stator slots does not affect the optimal number of rotor slits. The product of the slit
number and slit width seems to have an optimal value, which is a function of the rotor slip. As
the slip increases, the optimal product of the slit number and slit width increases. The amount of
air between the rotor steel teeth should be large enough in order to coerce the magnetic flux to
penetrate deeper into the rotor. When the rotor frequency increases, more air is needed in order
to force the flux to penetrate effectively. If the slits are too narrow, the flux crosses over the slits
in tangential direction and the penetration depth decreases. The slits should extend very deep
inside the rotor, but there should also be enough space for the magnetic flux to flow to the other
pole. The best results are attained when every second slit is deeper than the other.
138
5. Conclusion
5.1 Discussion
Three different calculation methods of evaluating the performance of solid-rotor induction
motors have been used in this thesis: the multi-layer transfer-matrix method (MTLM), the time-
harmonic FEM and time-stepping FEM.
First, the traditional multi-layer transfer-matrix method (MTLM) was further developed to
provide reliable information on practical solid-rotor design. This calculation method was
adapted, particularly, to the design of high-speed slitted solid-rotor induction motors.
Improvement of the multi-layer transfer-matrix method was achieved by introducing into the
method a new end-effect factor and a new curvature factor for slitted solid rotors equipped with
well-conducting end rings. These improvements do not, however, solve the calculation problem
scientifically unambiguously. Several simplifications have still been used. The slitted rotor
section was modelled by a non-isotropic 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. The results showed that if the
ratio of slit and tooth widths or slit depths become very low or large, the assumption may break
down. Calculation in polar-co-ordinates could improve the accuracy of the method. However,
this calculation method will remain a tool that does not evaluate exactly the behaviour of the
electromagnetic fields in a rotating three-dimensional solid rotor. The designer should always
use a more accurate method to critically evaluate the results.
Before the development of the MTLM method introduced here, the author could not see any
remarkable improvement lately done on the method. To design solid-rotor machines most
researchers rely completely on the numerical methods. Therefore, the solution for the MTLM
method demonstrated in this work may be considered innovative. The results achieved with the
MTLM coincide satisfyingly with the results got from the FEM and from the measurements.
In the numerical calculations the time-harmonic and time-stepping FEM were applied. The
author has no scientific contribution to the development of the time-harmonic or the time-
stepping methods used in the work. The time-harmonic calculation method is much faster to use
than the time-stepping method. Usually, present-day researchers favour the time-harmonic
139
method. In most of the cases, it satisfactorily describes the fundamental behaviour of an
induction machine. The method, however, has also considerable restrictions. It should be
emphasized to the user of the time-harmonic method that, in some cases, calculation might fail.
In a late phase of this study, defects in time-harmonic FEM-calculation were noticed. The
method may not be used straight in conjunction with all possible geometries. Especially with
respect to such geometries known for producing large torque ripple at positive rotating speeds,
it should be evaluated extremely carefully whether to use time-harmonic FEM-calculation.
In most of the cases, the time-harmonic finite element analysis was used to investigate the
problems concerning the slitted solid-rotor induction machine. In order to obtain correct results,
the researcher must carefully apply this method. The practical analysis of the slitted solid rotor
provides to the designer important guidelines. However, several problems still remain for
further research. As an example, it may be mentioned that the rotor fields are solved in two
dimensions only. Also, time-stepping calculating should be favoured in order to avoid errors,
which may occur in time-harmonic calculation results. Time-stepping calculation describes the
fields more accurately and, first of all, describes the phenomena movement of the rotor, which
is essential for the evaluation of the rotor losses.
Some studies on slitting patterns for solid rotors could be found, but the examinations were very
restricted; they were not done for high-speed machines, the parameter variation was done within
a very narrow range, the electromagnetically best slitting alternatives could be found but the
practical manufacturing conditions were disregarded. As a consequence, the results of this study
and the earlier studies are not easily comparable. There are also no examinations available
considering active length to diameter ratios for solid-rotor induction motors.
5.2 Future work
In future work, a full three-dimensional time-stepping FEM study should be performed in order
to obtain an accurate behaviour pattern of the electromagnetic phenomena in a solid rotor,
especially in the end regions. However, it may probably not be possible to carry out such an
investigation in the near future since a large increase of calculation power will be required.
Nevertheless, the methods introduced in this work seem to offer sufficient enough tools to the
experienced designer of solid-rotor induction machines.
140
The slitting of the solid rotor was only studied from the torque producing point of view. The
motor efficiency and the power factor should also be studied through an effective time transient
FEM solver.
Although the mechanic and thermodynamic features of a solid rotor machine were only
limitedly considered in this work, it could be of great interest to combine in future research
work all of these fascinating scientific fields into one calculation program. On the market, there
are already available commercial FEM programs, which are able to solve, separately, all of
these problems. However, solving even one of these problems by means of a present-day three-
dimensional FEM-program is, practically, still a too slow process and this means that
combining the different fields of problem in practical design will not be possible in the near
future. In consequence, there is still need for an analytical lumped parameter analysis.
5.3 Conclusions
This thesis summarises two decades’ experience Lappeenranta University of Technology has
achieved in the field of solid-rotor induction machines. As an example, the utilisation factors of
different solid-rotor machine types given in chapter 4 introduce unique information about the
design of solid-rotor machines. The knowledge is based on the wide experience in the field of
study and on the competence of constructing and measuring prototypes and also of delivering
motors for industrial use. The author had and still has a significant role in the above-mentioned
project work since he has been, in fact, the only post-graduate researcher at LUT active in the
field of electromagnetic design of solid-rotor machines during the latest ten years. The author
designed most of the machines mentioned in Fig. 4.1 and carried out most of the measurements
on the larger solid-rotor machines reported in Fig. 4.1.
The thesis studied assorted parts of high-speed motor technology. The study focused on solid-
steel rotors and their dimensioning. Special attention was given to the axially slitted solid-steel
rotor equipped with end rings made of a high-conductivity non-magnetic material such as
copper or aluminium. This rotor type proved to be useful for industrial applications running at
moderately high speeds and moderate power. Motors equipped with this rotor type are running
in industrial applications which are, for example, following: 200 kW, 8600 min-1 blowers, 300
kW, 10200 min-1 vacuum blowers, 250 kW, 9000 min-1 high pressure blowers, 1000 kW, 12000
min-1 compressor, 400 kW, 6000 min-1 aeration compressors etc. In recent years, it was the
responsibility of the author to perform the design of such machines. The solid-rotor technology
141
seems to be adaptable also to machines larger than those mentioned above and, because of that,
there clearly exists a need for fast design tools to be used for the design of such machines. It is
important to obtain a nearly final solution as fast as possible. In such a case, modern FEM-tools
may then be applied to achieve the final design.
Two main scientific tasks were selected:
The first task was selected to find out the competence of the analytical method published by
Pyrhönen (1991a) and then to further develop the method in order to create a practical and –
from the product development point of view – accurate enough calculation procedure for the
solid-rotor machines. This target includes both the electromagnetic field calculation problem of
the solid rotor and the loss calculation of this rotor type. The second paragraph of the thesis
studies the analytical calculation features of the solid rotor. The third chapter is devoted to the
losses of the solid-rotor machine. Special interest is focused on the rotor surface losses that are
minimized by using 5/6-short-pitch windings and by minimising the permeance harmonics of
the air-gap flux density at the solid-rotor surface. Different methods to minimise the losses are
studied and compared in detail. A new method to practically eliminate the permeance
harmonics is introduced – a specially formed semi-magnetic slot wedge.
The objective of the second task was to optimise the slitted solid-rotor construction by using
FEM-based computational tools. The analytical results achieved by the calculation method
developed were compared to the results achieved with the FEM and by means of laboratory
measurements. These studies and results are described in the fourth chapter of the thesis.
The two-dimensional multi-layer transfer-matrix method was chosen to fast evaluate the
electromagnetic fields in a solid rotor. Some important phenomena that could not be dealt with
by means of the selected method were solved by using substitute parameters for the rotor
material properties. These phenomena are the rotor end effects, rotor slitting and rotor
curvature. The results given by the method developed concur well with the measured and the
FEM results. Only when extreme alternatives are calculated the method is not accurate enough
for practical design purposes. Such conditions could be, for example, the number of rotor slits
that is very low or high, or slits or teeth that are extremely narrow.
It is the further aim of the second task that the results of this thesis should give guidelines for
slitting of the solid rotor. Slitting diminishes the rotor surface impedance to a considerable
142
degree. The slits coerce the flux to penetrate as deep as the slits are; hence, the effective rotor
resistance decreases considerably. In order to achieve the highest possible rotor torque, the most
appropriate solid-rotor surface axial slitting should be found. The number of the stator slots
does not affect the optimal number of the rotor slits and the product of the slit number and the
slit width seems to give an optimal value. The slits should reach as deep as possible, but there
should also be reserved enough space for the magnetic flux to flow into the other pole. It is
evident that also the mechanical aspects must be considered. A good rule of thumb, as for the
electromagnetic properties of the machine, is that the slit depth should be about half the rotor
radius.
143
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153
Appendix A
Test motor constructions
In the following, the solid-rotor calculation developed during the project of this thesis is used to
determine the operation points of the test motors, which were designed at LUT. Laboratory
measurements were made to confirm the calculation results. The rated powers of the test motors
are 200 - 400 kW, depending on the active length of the motor and on the rotor structure. There
are cooling blowers at both ends of the shaft. The power of the blowers is taken into account in
the efficiency measurements. The machine has a 5/6-short-pitch stator winding. The parameters
of the test machine are given in table A.1 and Fig. A.1 shows a schematic drawing of the test
machine.
Table A.1. The parameters of the 250 … 300 kW test machines.
Number of pole pairs, p 1 Stator voltage, US [V] 400 Connection Delta Rated frequency, fs [Hz] 140
Rotor active volume [cm3] 8000 Diameter ratio, L/D 1.4 – 1.7 Stator inner diameter [mm] 200 Stator outer diameter [mm] 400 Number of stator slots, QS 48 Number of turns in series per stator winding, NS 16 Number of rotor slits, QR 34 Stator material M250-50A Rotor material Fe-52C Resistivity of the rotor material @ 20 °C, ρ [µΩcm] 25.7
Temperature coefficient of the resistivity ε [1/K] 0.0115
154
rotor
roottorirunko/-uritus
stator laminations / windings
stator laminations
end-ring
Fig. A.1. A schematic drawing of the analysed high-speed solid-rotor machine.
The magnetic properties of the rotor material are given in Fig. A.2.
0
0,5
1
1,5
2
-1000 0 1000 2000 3000 4000 5000 6000 7000H [A/m]
B [T] x 10
Fig. A.2. BH-curve of the rotor material Fe-52C.
155
Appendix B
Measuring arrangement and measured characteristics of a test motor
The characteristics of the test machines were measured in the MOTORIUM CARELIAE
laboratory of Lappeenranta University of Technology. During the load tests the shafts of two
similar machines were mechanically connected via a torque measurement shaft. The
intermediate circuits of the supplying inverters were also connected together. Thus, only the
losses of the system were taken out from the grid. The measuring arrangement is shown in Fig.
B.1.
PWM
Poweranalyser
Torque and speed sensor
Solid-rotor generator
Solid-rotormotor
Oscilloscope
Temperature measurement
Torque and speedmeasurement
PC data acquisition
PWM
Frequency converter
Oscilloscope
IEEE-488 GPIB
Filter
power grid
Power flow
Frequency converter
Fig. B.1. The measuring arrangement of the test machines. The test machine was fed by a VACON 400
CX 5 -inverter. The torque and the rotating speed were measured with a VIBRO METER TG-
20 torque transducer. A YOKOGAWA PZ4000 or NORMA D 6100 AC-power analyser
equipped with three current transformers measured the electric power. The waveforms of the
156
supply are analysed with YOKOGAWA PZ4000 or TEKTRONIX THS 720 -digital
oscilloscope. Six thermo-elements were mounted in the stator windings for temperature
analyses. A FLUKE HYDRA 2620A measured the temperatures.
In the following, the measured results of a test machine of 250 kW, 8400 min-1 are given. The
machine L/D ratio is 1.7. The machines were tested with both a sinusoidal and an inverter
supply. The machines equipped with an inverter supply were also tested with a sine wave filter
installed between the inverter and the machine. Although an inverter supply is added to the
machine, the filter provides almost sinusoidal waveforms for the motor currents. Therefore, the
characteristics of the test machine may be assumed to be measured at sinusoidal supply. The
filter compensates the large reactive current of the machine, thus from the inverter side, as the
power factor is concerned, the solid-rotor machine resembles a similar laminated rotor machine.
The motor voltage waveforms are shown in Fig. B.2 with an inverter supply using 6 kHz
switching frequency and with a filtered supply.
-800
-600
-400
-200
0
200
400
600
800
-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.01
a) -800
-600
-400
-200
0
200
400
600
800
-1.0E
-03
-2.0E
-04
6.0E-04
1.4E-03
2.2E-03
3.0E-03
3.8E-03
4.6E-03
5.4E-03
6.2E-03
7.0E-03
7.8E-03
8.6E-03
9.8E-03
1.1E-02
1.3E-02
1.5E-02
1.6E-02
1.8E-02
1.9E-02
2.1E-02
2.3E-02
2.4E-02
2.6E-02
2.7E-02
Vol
tage
[V]
b)
Fig. B.2. Voltage waveforms of a 250 kW, 8400 min-1 solid-rotor motor with a) 6.0 kHz switching
frequency without filter and b) with a filtered output.
The motor current waveforms at 250 kW load are shown in Fig. B.3 with a frequency converter
supply using 3.6 kHz, 6 kHz and 12 kHz switching frequencies, and with a filtered supply.
157
-800
-600
-400
-200
0
200
400
600
800
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.0
Stat
or c
urre
nt [A
]
a) -800
-600
-400
-200
0
200
400
600
800
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02
Stat
or c
urre
nt [A
]
b)
-800
-600
-400
-200
0
200
400
600
800
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.0
Stat
or c
urre
nt [A
]
c) -800
-600
-400
-200
0
200
400
600
800
-1.0E
-03
-2.0E
-04
6.0E-04
1.4E-03
2.2E-03
3.0E-03
3.8E-03
4.6E-03
5.4E-03
6.2E-03
7.0E-03
7.8E-03
8.6E-03
9.8E-03
1.1E-02
1.3E-02
1.5E-02
1.6E-02
1.8E-02
1.9E-02
2.1E-02
2.3E-02
2.4E-02
2.6E-02
2.7E-02
Stat
or c
urre
nt [A
]
d) Fig. B.3. Loaded current waveforms of a 250 kW, 8400 min-1 solid rotor motor with different switching
frequencies without filter and with filtered output. A) The switching frequency is 3.6 kHz, b)
the switching frequency is 6 kHz, c) the switching frequency is 12 kHz and d) filtered output
current.
The no-load tests were measured using both non-filtered and filtered supplies. Since the
harmonics caused by a frequency converter produce a great amount of eddy current losses in the
surface of the solid-rotor steel, this test shows clearly the influence of the filter on the iron
losses. In Fig. B.4 the motor no-load losses are drawn as a function of the motor voltage at
different switching frequencies and at nominal output frequency. The mechanical losses are 3.1
kW, which include the friction losses and the power of the cooling system. These are 1.2 % of
the nominal power.
Although the solid steel rotor requires quite a large slip frequency to produce the torque, the
values of the relative slip remain small at elevated speeds. In this case, the nominal slip
frequency is 2.25 Hz. As Fig. B.5 shows, the loaded solid-rotor motor runs at a low relative slip.
Thus, the fundamental frequency losses in the rotor are low. Now, the nominal relative per-unit
slip is 1.6 % without filter and 1.9 % with filter.
158
0
2000
4000
6000
8000
10000
12000
0 50 100 150 200 250 300 350 400 450U [V]
P loss [W]
Switching frequency 3.6 kHz
Switching frequency 6.0 kHz
Switching frequency 12.0 kHz
Filtered supply
Fig. B.4. No-load losses of a 250 kW, 8400 min-1 solid-rotor motor with different switching frequencies
without filter and with filtered output.
0
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
s [%]
P mech [kW]
Shaft power with filter
Shaft power without filter
Fig. B.5. Motor output power as a function of the rotor slip at rated frequency.
159
The motor efficiency is drawn at nominal supply frequency in Fig. B.6. The filter considerably
improves the motor efficiency despite of the reduced power capacity caused by the voltage drop
in the filter. The maximum efficiency with the filtered supply is 95.5 %. Because of the filter
the efficiency improvement varies between 5...1.5 %-units as the power increases from 25 % to
100 % of the nominal load, respectively.
87
88
89
90
91
92
93
94
95
96
97
0 50 100 150 200 250 300P shaft [kW]
η [%]
With filter
Without filter
Fig. B.6. Efficiency as a function of the output power of a 250 kW, 8400 min-1 solid rotor motor without
filter at 6 kHz switching frequency and with filter.
Fig. B.7 a) points out the disadvantage of the solid-rotor induction motor. Without a cage-
winding in the rotor the motor power factor stays close to 0.7. This is a consequence of the fact
that the same rotor steel parts function as a path for both the fundamental rotor current and the
main magnetic flux. Thus, the angle of the rotor impedance is going to vary typically between
30° and 45°.
Characteristic for the solid-rotor machine is that the magnetic field strength at the surface of the
rotor is sufficiently high to drive well the rotor steel into magnetic saturation. It is for that
reason that the motor magnetizing current increases as the load is increased, this is shown in
Fig. B.7 b). Thus, the power factor does not rise to that level which is normally achieved in
moderate power two-pole squirrel-cage induction motors. The figure shows also the active
current of the motor. The motor current and the inverter output current are shown in Fig. B.7 c).
160
The LC-filter is designed so that it compensates about 80 % of the no-load magnetizing current
of the motor. As a consequence, the inverter output current is about 20 % lower than the motor
current at nominal load. Therefore, the inverter of the solid-rotor induction motor drive does not
need over-sizing despite of the poor power factor and the same inverter can drive both a solid-
rotor induction motor and a laminated rotor induction motor of the same power.
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 50 100 150 200 250P shaft [kW]
cos φmotor power factor
power factor from inverter output
a)
0
50
100
150
200
250
300
350
400
450
0 50 100 150 200 250 300
P shaft [kW]
Cur
rent
[A]
Active current
Reactive current
b)
161
0
100
200
300
400
500
600
0 50 100 150 200 250P mec [kW]
I [A] Motor current
Inverter output current
c)
Fig. B.7. a) Power factor as a function of the output power of a 250 kW, 8400 min-1 solid-rotor motor
and of a solid-rotor motor with a sine-filter. b) Active and reactive current of a 250 kW, 8400
min-1 solid-rotor motor. c) Motor current and inverter output current as a function of the output
power of a 250 kW, 8400 min-1 solid-rotor motor.
162
Appendix C
Eddy Currents and the displacement current term
Varying magnetic fields induce time-dependent eddy currents in conducting materials. In this
case, we are interested in iron cores that have a large permeability µ and large conductivity σ.
In a conducting material the penetration of an external magnetic flux creates also an internal
eddy current that, again, creates its own magnetic field, and so on.
The current density J depends on the conductivity σ and the electric field strength E as J = σE.
Now, the induction law (2.5) may be written as
HBJtt ∂
∂σµ∂∂σ −=−=×∇ . (C.1)
The full Ampère law (2.4) is inserted into the left side of (C.1) and we get
HHtt
D∂∂σµ−=
∂∂
−×∇×∇ . (C.2)
Since ( ) ( ) HHH 2∇−⋅∇∇=×∇×∇ and D = εE we get
02
22 =−−∇ HHH
tt ∂∂µε
∂∂σµ . (C.3)
When slowly changing phenomena, e.g. in metals, are examined the last term in the equation is
small and may be left out and we get the diffusion equation for the eddy currents. If, instead, the
medium is a pure insulator the term in the middle is zero and we get a wave equation as a result.
Which one of the time derivative terms in Eq. (C.3) is more important depends on the material
parameters and the angular frequency of the phenomenon. If
163
εωσ
>> , (C.4)
the fields behave according to the diffusion equation. If the conductivity does not have a large
effect on the phenomenon
εωσ
<< , (C.5)
and the wave equation is selected.
In solid rotors the conductivity is large and the angular velocities are small and thus the
diffusion equation is selected. This means that the displacement term in (2.4) may be left out
accordingly. (Sihvola 1996)
164
Appendix D
Two-dimensional eddy current problems
Substituting the magnetic vector potential definition in the induction law we get
AEt∂
∂×−∇=×∇ (D.1)
The electric field strength may be written as
φ∂∂
∇−−=tAE (D.2)
where φ is the reduced electric scalar potential. Eq. (D.2) fulfils Eq. (D.1) since 0≡∇×∇ φ .
The induction law is now automatically satisfied. Eq. (D.2) describes how the electric field is
composed of two different parts, one rotational part is induced by the time variation of the
magnetic field and the non-rotational part is created by the electric charges and polarization of
dielectric materials.
The current density may be written as
φσ∂∂σσ ∇−−==
tAEJ . (D.3)
Ampère’s law and the determination of the vector potential give
JA =
×∇×∇
µ1 . (D.4)
Substituting (D.3) to (D.4) gives
01=∇++
×∇×∇ φσ
∂∂σ
µ tAA . (D.5)
165
Eq (D.5) is valid in the eddy-current regions while (D.4) is valid in the regions of the source
currents J = Js − such as the coil currents – and in the regions with no current densities at all J =
0.
Often, a two-dimensional solution is used in electric machinery and in such cases the solution
may be based on one single component of the vector potential A. The field solution (B, H) is
found in the xy-plane while J, A, and E have only the z-component. The gradient φ∇ has only
a z-component too, since J and A are in z-direction and (D.3) is valid. The reduced scalar
potential is thus independent of the x- and y-components. φ could be a linear function of the z-
coordinate, since the two-dimensional field solution does not vary as a function of z. The
assumption of two-dimensionality is no longer valid if there exist potential differences created
by electric charges or polarization of dielectric materials. In two-dimensional eddy current
problems the scalar potential must be set 0=φ .
In a two-dimensional case (D.5) becomes
01=+
∇⋅∇−
tAA z
z ∂∂σ
µ. (D.6)
Outside the eddy current regions
z1 JAz =
∇⋅∇−
µ (D.7)
must be used. (Silvester 1990).
166
Appendix E
Poynting’s theorem
The equation that will lead to a power equation may be derived from (2.4) and (2.5) by dot-
multiplying (2.4) with E and then subtracting it from (2.5) after the latter is dot-multiplied with
H. Using the vector identity )( HEHEEH ×⋅∇=×∇⋅−×∇⋅ , we find
EJDEBHHE ⋅−∂∂
⋅−∂∂
⋅−=×⋅∇tt
)( (E.1)
Using the constitutive relations (2.8) and (2.9) and noticing that
⋅
∂∂
=∂
∂⋅ HHHH
21
tt
021
21)( =⋅+
⋅
∂∂
+
⋅
∂∂
+×⋅∇ EJEEHHHE εµtt
. (E.2)
Each term in the above equation has the unit watts/m3. The term EJ ⋅ is recognized as the
ohmic loss Pc. The term HH ⋅µ)2/1( is identified as the stored magnetic energy Wm per unit
volume and the term EE ⋅ε)2/1( is identified as the stored electric energy We per unit volume.
Integrating (E.2) over an arbitrary volume V the power is obtained,
∫ ∫∫∫ ⋅−
⋅+
⋅
∂∂
−=×=×⋅∇V VVS
vvt
v dd21
21d)(d)( EJEEHHsHEHE εµ , (E.3)
where the surface S encloses the volume V. This is known as the Poynting theorem. It tells
about the power balance inside a volume V of the electromagnetic field.
The instantaneous Poynting Vector is defined as
HES ×= .
167
The unit of S is watts/m2, which is the power density on a surface. The first term in (E.2) is
S⋅∇ . Remembering that the divergence of a vector represents the outflow of the vector across
a small volume, S⋅∇ is identified as the outflow of the electromagnetic power. Equation (E.2)
is a statement of the conversation of power. It states that the sum of the electromagnetic power
flowing out of the volume, the rate of the increase of stored magnetic and electric energy in that
volume, and the power lost to ohmic heat must equal zero.
If the permittivity and the permeability are assumed constants, Poynting’s theorem for time-
harmonic vectors is
∫ ∫∫∫ ⋅−
⋅+
⋅−=×=×⋅∇
V VVS
vvjv d21d
41
412d)(
21d)(
21 ***** EJEEHHsHEHE εµω ,
(E.4)
and the complex Poynting vector is
*
21' HES ×= . (E.5)
The complex Poynting vector contains both reactive and real power parts, which, in this case,
correspond to the real and reactive power flow into the rotor. If we calculate the apparent power
of the rotor by integrating the sentence
× *
21 HE over the rotor surfaces we get the apparent
power flow into the rotor. This apparent power phase angle corresponds to the phase angle of
the rotor impedance.
The instantaneous Poynting vector can be written in terms of the phasors
[ ] [ ]tω2j** eRe21Re
21 HEHES ×+×= (E.6)
For the time-harmonic electromagnetic fields, the time-average Poynting’s vector Save is defined
as the average of the time-domain Poynting vector S(x,y,z,t) over a period T = 2π/ω. Since the
term [ ]tω2j*eHE × vanishes, the time-average Poynting vector is given by
168
[ ]*ave Re
21 HES ×= . (E.7)
This formula is a useful tool for computing the time-average electromagnetic power flow. (Shen 1995).
