7373941 010-high-speed-solid-rotor-induction-motor


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7373941 010-high-speed-solid-rotor-induction-motor

  1. 1. 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
  2. 2. ISBN 951-764-981-9 ISBN 951-764-944-4 (PDF) ISSN 1456-4491 Lappeenrannan teknillinen yliopisto Digipaino 2004
  3. 3. 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
  4. 4. 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
  5. 5. Acknowledgements In 1996, at the Laboratory of Electrical Engineering, Lappeenranta University of Technology, the research activities related to this thesis got started, being part of the project “Development of 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
  6. 6. 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
  7. 7. 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
  8. 8. 9 Abbreviations and symbols Roman letters a abbreviation, function, number of parallel conductors, constant a1k factor for calculating the slot harmonic amplitudes A area, linear current density, vector potential Aj 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
  9. 9. 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
  10. 10. 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 τp pole pitch τu slot pitch Φ magnetic flux χ chord factor ωs stator angular frequency Ω mechanical rotating angular speed
  11. 11. 12 Subscripts ave average c cylindrical shell region, conductor C Carter Cu copper class classical dyn dynamic e electric ec eddy current em electromagnetic er end region exc excess Fe iron fr friction i index in input harm harmonic hys hysteresis k layer lin linear m magnetic max maximum value mech mechanical min minimum value R rotor s supply, synchronous S stator sl slip sw switching t tooth tot total u slot, slit v harmonic of order v x, y, z coordinates δ air-gap 0 basic value, initial value 1 fundamental, bottom layer
  12. 12. 13 Superscripts R rotor S stator Other notations a magnitude of a a complex form of a a vector a (in x, y, z coordinates) a complex form of vector a (time-harmonic presentation) aˆ peak value of a Acronyms AC alternating current emf electromotive force DC direct current FEM finite element method IGBT insulated gate bipolar transistor IM induction machine MLTM multi-layer transfer-matrix mmf magnetomotive force PMSM permanent magnet synchronous machine PWM pulse width modulation SM synchronous machine
  13. 13. 14
  14. 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
  15. 15. 16 that run at moderate speeds and with moderate power. The motor supply frequencies vary between 100 Hz and 300 Hz and the motor powers between 100 ... 1000 kW. The idea of using 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.
  16. 16. 17 The laboratory of electrical engineering at Lappeenranta University of Technology (LUT) has an over two decades long experience in and knowledge about the design and manufacturing of 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.
  17. 17. 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
  18. 18. 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] Maximumpower[kW] Laminated rotor Solid rotor Fig. 1.1. Powers limited by the rotor material yield stress (700 MPa) versus rotational speed (Jokinen 1988). High-speed machines are mainly applied to blowers, fans, compressors, pumps, turbines and spindle machines. The best efficiencies for these devices are achieved at elevated speeds, and by using high-speed machines gearboxes and couplings can be avoided. The biggest potential for high-speed machines lies on the field of turbo-machinery. Potential applications are blowers, fans, gas compressors and gas turbines, because the rotational speeds of the gas compression units are typically high. A common way to manufacture a gas compression unit is to use a standard electric motor and a speed-increasing gearbox. Such machinery is manufactured by Atlas Copco, Dresser-Rand, Solar Turbines, MAN Turbo, etc. During the latest decades high- speed machines have been pushed on the market as an interesting solution to increase the total system efficiency and to minimise total costs.
  19. 19. 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
  20. 20. 21 motor type is commonly used and in smaller power ranges also the permanent magnet machine type is used. Another conclusion might be that large natural gas pumping 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
  21. 21. 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 f f f ff Ω Ω Ω ΩΩ s = − == − == − = ω ω ω ωω . (1.1)
  22. 22. 23 Here, Ωsl describes the mechanical angular slip speed of the rotor, ωsl the electrical slip angular speed of the rotor and fsl the electrical slip frequency in the rotor. In motoring mode the slip s is positive and in generating mode the slip is negative. The relation between the angular speeds, pole pair number p, torque and power may be written as em R em R emRe π2 T p f T p TΩP === ω (1.2) The slip frequency fsl and the slip angular speed ωsl in the rotor are of great importance, especially in 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 I2 R 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
  23. 23. 24 profitable because by milling axial slits into the rotor it is possible to get considerably more power, a slightly better power factor and a higher efficiency than it may be achieved with a smooth rotor, and the machining costs remain moderate. Rotor coating, end rings and squirrel- cage structures raise the manufacturing demands and costs, but these structures boost the motor torque and properties in a considerable way. For example, according to the experience of the author, a smooth solid rotor equipped with copper end rings produces twice as much torque at a certain slip as the same rotor without end rings and a motor with a 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).
  24. 24. 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)
  25. 25. 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.
  26. 26. 27 • Integrated machines. The rotating part of the load machinery forms the rotor, for example conveyer idle, where the stator can be outside or inside of the rotor. • 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.
  27. 27. 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:
  28. 28. 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).
  29. 29. 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.
  30. 30. 31 2. Solution of the electromagnetic fields in a solid rotor This chapter describes the development and gives a review of the analytical methods that have been introduced for the solving of the electromagnetic fields in 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
  31. 31. 32 analytical solution for the rotor fields has to be found. The ultimate simplification is to solve the Maxwell’s field equations assuming a smooth rotor and a magnetically linear rotor material. The literature in the field widely deals with the analysis of the solid rotor, especially in the 1950’s, 1960’s and 1970’s. Research was carried out with the objective to maximize the starting torque and to minimize the starting current and, further, to simplify the rotor construction of an induction machine. In the articles it is commonly supposed that the rotor is infinitely long. Another assumption made is that the rotor material is magnetically linear or the rotor material has an ideal rectangular 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),
  32. 32. 33 Dorairaj (1967b), Freeman (1968), Sarma (1972), Chalmers (1984), and Sharma (1996). Cullen (1958) used the concept of wave impedance. To define the impedance of the solid rotor a 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).
  33. 33. 34 The method described above was later developed by Freeman (1970) who published a new version on the technique used for polar coordinates. This technique was also used by Riepe (1981b). Yamada (1970), Chalmers (1982) and Bergmann (1982) used the MLTM method in the Cartesian coordinates. 2.1 Solution of the electromagnetic rotor fields under constant permeability In the following analysis, a field solution is derived for a linearized, smooth rotor of finite length. The solution is written in the form of a 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 p r tsxx s RS )1( ω−−= , (2.1)
  34. 34. 35 where p is number of pole pairs, r is rotor radius, t is time and ωs is stator angular speed. y x z Fig. 2.1. Coordinate system at the surface of the rotor. The next abbreviation is taken into use. The constant a is dependent on the dimensions of the machine p a τ π = , where τp is pole pitch, p D p 2 π =τ . (2.2) Equation (2.1) can be rewritten now tsaxtax s R s S ωω +=+ (2.3) Henceforth, the superscript R, which indicates to coordinate fixed to the rotor, will be left out. The differential forms of Maxwell’s equations have to be used as a starting point. Ampere’s law relates the magnetic field strength H with the electrical current density J and the electric flux density D. Faraday’s induction law determines the connection between the electric field strength E and the magnetic flux density B. Gauss’ equations definitely reveal that the divergence of B is zero and the divergence of D is charge density ρ, i.e. B has no source and D has the source and the drain. t∂ ∂ D JH +=×∇ , ( 2.4)
  35. 35. 36 t∂ ∂ B E −=×∇ , (2.5) 0=⋅∇ B , (2.6) ρ=⋅∇ D , (2.7) The latter part of equation (2.4) representing Maxwell’s displacement current is omitted, because the problem is assumed to be 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
  36. 36. 37 0=⋅∇ A . (2.12) When equation (2.11) is substituted to Faraday’s law equation (2.5) we get 0=      ∂ ∂ +×∇ A t E . (2.13) The sentence in parenthesis has no curl and may thus be written as a gradient of a scalar function −φ. Now, the electric field strength can be written in the following form φ ∂ ∂ ∇−−= t A E . (2.14) The charge density ρ can be assumed to be negligible in 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∂ ∂ −=∇−⋅∇∇=×∇×∇ A AAA µσ2 )()( . (2.15) When the Cartesian coordinates are used and the Coulomb gauge, equation (2.12), is valid, the differential equation of A can be expressed by t A z A y A x A iiii ∂ ∂ µσ ∂ ∂ ∂ ∂ ∂ ∂ =++ 2 2 2 2 2 2 , (2.16) where i is x, y, or z (Yee 1971). Because all fields in the induction machine may be assumed to vary sinusoidally as a function of time, a steady state 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
  37. 37. 38 [ ]ts zyxtzyx sj e),,(Re),,,( ω AA = , (2.17) where A is a complex and only position dependent vector. The space structure of the stator winding of the induction machine causes the vector potential A to vary in the direction of the x- axis both as a function of place x with the term ejax and as a function of time t with the term ej ss tω . The vector potential is obtained in form of a complex vector function [ ])(j s e),(Re),,,( tsax zytzyx ω+ = AA . (2.18) Now, equation (2.16) can be written as a complex exponent function i ii Aa z A y A )( 22 2 2 2 2 λ ∂ ∂ ∂ ∂ +=+ , (2.19) where p s j2 j d s == µσωλ , (2.20) dp is the penetration depth and λ describes the wave penetration to a medium. The equations (2.16) - (2.19) can be written analytically as phasor equations. For instance, equation (2.4) in a time harmonic form is DJH ωj+=×∇ . (2.21) Using the annotation γ, which describes the variation of the fields in the axial direction, and δ for the air-gap length we get r 2 δµ λ γ += a . (2.22) Pyrhönen (1991a) repeated a mathematical deduction to the solution, which is convergent to the solution given by Yee (1971). In deriving the solution for the rotor fields the necessary boundary conditions to the solution are chosen in a convenient manner as: 1. The current has no axial component at the ends of the rotor.
  38. 38. 39 2. The magnetic flux density has no axial component at the ends of the rotor. 3. All field quantities disappear, when y approaches -∞, because the flux penetrates into the conducting material and attenuates. 4. The machine is symmetrical in 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 s e ) 2 sinh( )sinh( )ee( ) 2 sinh( )sinh( e tsaxyayy x L z L z GA ωλλ λ λ γ γ +           −+= , (2.23) )(j s e ) 2 sinh( )sinh( )ee(j tsaxayy y L z GA ωλ λ λ + −= , (2.24) )(j s e ) 2 sinh( )cosh( ) 2 coth() 2 coth(ej tsaxy z L zaLaL GA ωλ γ γ γ γ γ λ +           −+= , (2.25) where         ++ − = ) 2 coth() 2 coth()( ˆj r 2 00S LaL a KI G γ γ λ µ λ δ µ , Na p m K ξ π 0 = . (2.26) In the rotor the magnetic flux density equations are: ( ) )(j s e ) 2 sinh( )cosh( ee ) 2 sinh( )cosh( ) 2 coth() 2 coth(ej tsaxyayy x L z L zaLaL GB ωλλ λ λ γ γ γ γ γ λλ +             −+           −+= , (2.27)
  39. 39. 40 ( ) )(j s e ) 2 sinh( )cosh( ee ) 2 sinh( )cosh( ) 2 coth() 2 coth(e tsaxyayy y L z aL za a LaL GaB ωλλ λ λλ γ γ γ γ γ γ λ +             −+                   −++= , (2.28) )(j s e ) 2 sinh( )sinh( ) 2 sinh( )sinh( e tsaxy z L z L z GB ωλ λ λ γ γ λ +           −−= . (2.29) The tangential and the axial magnetic flux components per unit width on the surface of the rotor are found by integrating the respective flux densities: )(j 0 s e ) 2 sinh( )cosh( 1 ) 2 sinh( )cosh( ) 2 coth() 2 coth(jd tsax xx L z aL zaLaL GyBΦ ω λ λλ γ γ γ γ γ λ + ∞−                 −+−+== ∫ , (2.30) )(j 0 s e ) 2 sinh( )sinh( ) 2 sinh( )sinh( d tsax zz L z L z GyBΦ ω λ λ γ γ + ∞−           −−== ∫ . (2.31) The preceding field equations with respect to z are shown graphically in Fig. 2.2. As it is illustrated in the figure, Az and Hz are not zeros at the ends of the rotor, as it was required by the boundary conditions. This is a result of the approximations made to obtain the solutions. The dotted line sketches the forms of the actual distributions.
  40. 40. 41 AZ Ax HZ ΦZ Hx Φx 1 0 L / 2 L / 2 L / 2 1 0 1 0 Fig. 2.2. The axial distribution of the rotor fields at the surface of the rotor at standstill. The quantities are normalized with respect to the Az, Hx and Φx values at z = 0 (Yee 1971). a) Magnetic vector potential at y = 0, b) magnetic field strength at y = 0, c) magnetic flux 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 s e'Re tax JJ ω+ = , (2.32) lies between any two layers. Regions 1…K are layers made of material with resistivity ρk and relative permeability µk. The problem is to determine the field distribution in all regions, and hence, if required, the power loss in and forces acting on any region.
  41. 41. 42 K B H ρ µK K K-1 K-1 K-1 B H ρ µK-1 K-1 K-2 K-2 B Hk+1 k+1 B H ρ µ k k k B H ρ µk k k-1 k-1 ρ µk-1 k-1 y x z H -J'k 3 B H ρ µ3 3 2 2 2 B H ρ µ2 2 1 1 1 ρ µ1 1 . . . . . y = g 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
  42. 42. 43 medium are considered to be negligible. The current sheet varies sinusoidally in the x direction and with time; it is of infinite extent in the x direction, and of finite thickness in the y direction. Maxwell’s equations may be solved when the boundary conditions are as follows: (Greig 1967) 1. By is continuous across a boundary. 2. All field components disappear at y = ±∞ . 3. If a current sheet exists between two regions, then '1 JHH kk −= − . region k + 1 B H ρ µ k+1 k+1 k k region k B H ρ µ k k k-1 k-1 y = gk region k - 1 y = gk-1 d k ωk Fig. 2.4. Definition of the properties and dimension of region k (Freeman 1968). The following matrix equation may be written for region k, according to Greig (1967): [ ]       =                =      − − − − 1, 1, 1, 1, , , )cosh()sinh( )sinh( 1 )cosh( kx ky k kx ky kkkkk kk k kk kx ky H B H B dd dd H B T ΥΥβ Υ β Υ , (2.33) where k k k a µµ Υ β 0j = and kkkk sa σµµωΥ 0s 2 j+= (2.34) and [Tk], following Pipes (1956), is the transfer matrix for the region k. In the top region on the boundary gK 1,1, −− −= KyKKx BH β . (2.35) In the top region K the magnetic flux density and the magnetic field strength have to vanish gradually to zero according to boundary condition (2), thus (Greig 1967) )( 1,, 1 e yg KyKy KK BB − − − = Υ , (2.36)
  43. 43. 44 )( 1,, 1 e yg KxKKx KK HH − − − −= Υ β . (2.37) Solving the field in the bottom region on the boundary g1 1,11, yx BH β= . (2.38) In the region 1 the magnetic flux density and magnetic field strength must approach zero as y diminishes, it can be written (Greig 1967) )( 1,1, 11 e gy yy BB − = Υ , (2.39) )( 1,1, 11 e gy xx HH − = Υ . (2.40) The transfer matrix can be used as follows, considering the boundary conditions (1) and (3). The current sheet lies between regions k and (k+1). (Greig 1967). [ ][ ] [ ]       =      − 1, 1, 21 , , x y kk kx ky H B H B TTT L , (2.41) [ ][ ] [ ]       − =      +−− − − ', , 121 1, 1, JH B H B kx ky kKK Kx Ky TTT L . (2.42) The analysis above may be programmed to compute the electromagnetic fields and power flow at all boundaries. The computing can be initiated by using a presumed low value of the tangential field strength Hx,1 at the inner rotor boundary. The transfer matrix technique then evaluates By,k and Hx,k at all 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)
  44. 44. 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: ky k kz B a E ,, ω −= . (2.45) The net power density in a region is the difference between the power density in and the power density out (Greig 1967): ( )      −= −− * 1,1, * ,, s 2 Re kxkykxkyk HBHB a P ω . (2.46) The mechanical power density evolved by the region under slip sk is (Greig 1967) )1(.mech kkk sPP −= . (2.47) The ohmic loss I2 R elaborated by the region is (Greig 1967) kkkk PsPP =− ,mech . (2.48) 2.2.1 Definition of the fundamental permeability in a 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 1 ˆB of the Fourier series of the flux density is obtained by a numerical integration:
  45. 45. 46 ∫= π 0 1 )(d)sin()( π 2ˆ tttBB ωωω . (2.49) The fundamental permeability of a particular working point is defined as H B H ˆ ˆ )ˆ( 1 1 =µ . (2.50) B H H ω t H(ω t) B (ω t) B(ω t) 1 B1 ω t B H Fig. 2.5. The definition of the fundamental magnetic flux density B1(ωt) produced by an external impressed sinusoidally alternating magnetic field strength H(ωt) and the 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 H xHI p Nm xax x p j 2 de' 2π 4 2 R 0 j RR1 === ∫−τ ξ Θ , (2.51)
  46. 46. 47 from which the rotor current referred to the stator is found: xH Nam p I RR 2 jπ ' 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: y ax y B a LN xLB N U p p R s 2 2 - j RsR 2 2 jde 2 j' ξωξ ω τ τ −=−= ∫ . (2.53) Finally, the rotor impedance referred to the stator is found: x y H B p mLN I U Z R R 2 s R R R π )(2 ' ' ' ξω == . (2.54) 2.3 Effects of axial slits in a solid rotor The performance of an induction machine with a 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)
  47. 47. 48 investigated optimal rotor slitting, but his treatment of the subject is not expansive enough either. A slitted rotor may be solved by means of the MLTM method using substitute parameters for the permeability and the conductivity of the rotor material in the slitted region. The substitute parameters are obtained using a slit pitch τu, a slit and a tooth width wu and wt, relative permeability of the tooth µt and both slit and tooth resistivity ρu ja ρt, Fig. 2.6 (Freeman 1968). Here, it is assumed that the slit is not of a magnetic medium, i.e. µu = 1. The method considers the slitted rotor region to be replaceable by an equivalent homogenous but anisotropic medium. This assumption, however, leads to a solution, where the field distribution in slits and teeth regions would be equal. This, in fact, is far from reality, and thus the assumption should be considered carefully. If the slit geometry becomes more complicated, compared to the rectangular shapes, or if the wavelength of the travelling wave is small compared to the slit pitch, the assumption may break down. Possible skewing may not be taken into consideration. The substitute parameters are: u u u t t ττ µµ ww y += , (2.55) tut ut µ τµ µ ww x + = , (2.56) tuut utu ww ρρ τρρ ρ + = . (2.57) wt wu τu y x z Fig. 2.6. Slitted solid-rotor surface.
  48. 48. 49 2.4 End effects of the finite length solid rotor In the previous study the rotor was presumed to have an infinite length. Now, the effects of the finite length are considered. The problem of the end effects in solid rotors causes an indisputable difficulty. Several of the authors earlier mentioned did not take these effects into consideration at all. Omitting the problem may be justifiable if the rotor is equipped with thick end rings which have very low impedance and which make the current paths nearly axial. However, this supposition is not valid even in solid rotors with copper end rings because according to the experience of the author, when a solid rotor with copper end rings is used and the end effects are not considered, the calculated results give a 10 - 30 percent better torque at the given slip compared to measured results. Kesavamurthy (1959) introduced an empirical factor to modify the value of the rotor conductivity to incorporate the correction for the end effects. The author does not explain how the empirical factor for the end effect correction is achieved. Russel (1958) assumed that the rotor current density is confined in a thin shell around the rotor. Also Rajagopalan (1969) used this assumption. Jamieson (1968a) introduced the analysis in which the eddy currents are assumed to continue in the body of the rotor. He gives an equation for a correction factor of the end effects. Wood (1960c) made in his analysis a certain approximation, the validity of which is questioned. Angst (1962) proposed a complex factor that is applicable to the effective rotor impedance. Deriving the factor involves the solution of the 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
  49. 49. 50 equivalent conductivity takes the resistivity of the end rings into account when the rotor conductivity is considered. This technique was studied by Russell (1958), Jamieson (1968a), Rajagopalan (1969), Yee (1971), Woolley (1973), and Jinning (1987). The leakage inductance of the end rings can be ignored as infinitesimal. In other words, the rotor is analysed as being infinitely long, and the resistivity of the end rings is added to the resistivity of the rotor core steel. The analyses obtained by this method are very congruent to the measured results. Russell (1958) suggested that the actual loss in the rotor surface shell could be evaluated by assuming all the currents to be axial, but that the resistivity of the shell is increased by a factor ) 2 π tanh( π 2 1 1 p p L L τ τ α − = . (2.58) Further based on this, a general 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 R 1 2 11er )tanh(4 2 1                 ++= D pL kQQK , (2.60) where )tanh()(1 R R 1 R R 1 D pL k pL D Q +−= . (2.61) where erc cer 1 ρ ρ t t k = , and ter and ρer represent the end region effective thickness and the resistivity and tc and ρc represent the cylindrical shell region effective thickness and resistivity, respectively. If the rotor is slitted, the slit depth can be used for tc, otherwise an appropriate value for tc seems to be the depth of the flux penetration δp in the surface of the rotor. If the end
  50. 50. 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 R tR A L R σ = . (2.62) The resistance of the end ring in a tooth pitch is by the average diameter of the end ring Der, the conductivity of the end ring σer, the cross area of the end ring Aer and the number of the rotor teeth QR Rerer er er π QA D R σ = . (2.63) When a tooth current is marked as IsR, the end-ring current is (Richter, 1954)       = R sR er π sin2 Q p I I . (2.64) The currents cause copper losses in a rotor )2( 2 erer 2 sRtRRRCu, IRIRQP += . (2.65) In a two-pole rotor the number of phases is equal to the number of teeth, thus the resistance of the rotor phase is
  51. 51. 52       += R 2 er tRR π sin2 Q p R RR . (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 sR er R R K = . (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 er 2 2 coth 2 coth 2 1)( γ γ γ λ L aLaL aLsK −        +      += . (2.68) This factor takes also the loading into account. Ker(s) is analogous to the 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,
  52. 52. 53 2 2 coth1 2 coth1 er −            +             + = La aL La aL K . (2.69) Another theory proposed for the calculating of the end effects in a 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.
  53. 53. 54 The equations (2.23) - (2.29) give the rotor fields in rotor coordinates when a constant magnetic permeability is assumed. The rotor fields Ex(y=0), Ey(z=±L/2), Ex(z=±L/2), Hz(y=0), Hy(z=±L/2), Hx(z=±L/2) associated with Φ2 are defined directly from these equations since the flux Φ2 follows the unsaturated parts of the rotor. Ez(y=0) and Φ1 are defined from Hx(y=0) assuming that the magnetic properties of the material can be described using the 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. λ µ GH 1 0 = . (2.70) In addition, the following algebraic approximations are made as the loss of accuracy is negligible: a>>λ and γλ >> . Using equations (2.27) – (2.29) for the flux densities also gives the magnetic field strengths. Notifying that the phase angle of the imaginary unit is π/2 and the phase angle of the λ is π/4, it can be written (Pyrhönen 1991a):           −+= + = ) 2 sinh( )cosh( ) 2 coth() 2 coth(ee )(j4 π3 j 00 s1 L zaLaL HH tax yx γ γ γ γ γ λω , (2.71)           −−= + = ) 2 sinh( )sinh( ) 2 sinh( )sinh( ee )(j4 π j 00 s1 L z L z HH tax yz λ λ γ γω , (2.72) ) 2 coth(eee )(j4 π3 j 0 2 s1 L HH aytax L zx λω+ = = , (2.73) aytax L zy HH eee )(j4 π j 0 2 s1 ω+ = = . (2.74)
  54. 54. 55 The respective electric field strengths just outside the rotor surface are found by deriving equations (2.24) – (2.26) and by substituting the values of y and z: (Pyrhönen 1991a) ) 2 sinh( )sinh( e/e )(j lin0s 2 π j 00 s1 L z sHE tax yx γ γ ρµµω ω+ = −= , (2.75) )(j lin0s 2 π j 0 2 s1 ee/e taxay L zx sHE ω ρµµω + = −= , (2.76) )(j lin0s0 2 s1 ee/ taxay L zy sHE ω ρµµω + = −= . (2.77) The saturated components Hx(y=0) and Ez(y=0) are defined by the 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 R s 0 yyz B a s E ω == . (2.78) Φ2 can be obtained by integrating, over the surface y=0, that component of By(y=0) which corresponds to the tangential electric field strength Ex(y=0). The curl equation of the electric field strength gives t B z E x E yxz ∂ ∂ = ∂ ∂ − ∂ ∂ . (2.79) By choosing only the component that corresponds to the flux Φ2 equations (2.75) and (2.79) give ∫ + = ∂ ∂ = ) 2 sinh( )cosh( e/ed )(j 0 2 π j 0 1 2 L z s j H t z E B tax lins s Φ s γ γ γρµµω ω ω . (2.80) By integrating over the surface y = 0, the unsaturated path flux Φ2 is obtained as
  55. 55. 56 sH a xzBΦ p p L L Φ / 4 dd lin0s0 s 2 2 2 2 2 2 ρµµω ω τ τ == ∫ ∫ − − . (2.81) The air-gap voltage of the machine is calculated with Faraday’s induction law )( 2 j 21s ΦΦ N U +−= ξ ωδ . (2.82) By using the complex Poynting vector, see App. E, the average power density flow into the surface can be defined as * 2 1 HES ×= . (2.83) The complex power that flows into the rotor is found by integrating the Poynting vector over all the rotor surfaces (Yee 1972). By using equations (2.71) – (2.77), we obtain .d)( 2 1 d)( 2 1 d)( 2 1 d)( 2 1 π' 0 - 0 - 2/ * 2/2/ * 2/ 2/ 2/ 2/ 2/ 0 * 00 * 0     −+     +−= ∫ ∫ ∫ ∫ ∞ ∞ ==== − − ==== yHEyHE zHEzHED Lz yLzx Lz xLzy L L L L y zyx y xyzS (2.84) In equation (2.84) the field variations in the direction of the 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).
  56. 56. 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 R C, 2 1 D gD K k k − −= . (2.85) The slit pitch in the calculation region k is uC,u,' ττ ⋅= kk K , (2.86) and the tooth width is uuC,t,' wKw kk −⋅= τ . (2.87) Now, the equations (2.55) - (2.57) may be rewritten, as the curvature is taken into consideration.
  57. 57. 58 kk k ww k u, u u, t, ty '' ' )( ττ µµ += , (2.88) tut, u,t x ' ' )( µ τµ µ ww k k k + = , (2.89) k k ww k t,uut u,tu ' ' )( ρρ τρρ ρ + = . (2.90) g Fig. 2.8. Effects of the curvature to the 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. K K-1 k+1 k k-1 2 1 HK-1HK Hk+1 Hk Hk-1 H2 H1 r r r r r rK-1 k+1 k k-1 2 1 E E E E E E 1 2 k-1 k k+1 K-1 BK-1 B2 B1 B B B k-1 k k+1 r z Fig. 2.9. The cross-section through a K-layer cylindrical induction device.
  58. 58. 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 ky k kx ky kk kk kx ky H B H B db ca H B T , (2.91) where { })(')()(')( 12121 βββββ νννν IKKIak −−= , (2.92) { })()()()( 12121 ββββσ νννν KIIKrb kkk −−= − , (2.93) { })()()()(j 1212,1 ββββωµ ννννφ KIIKrc kkk −−= − , (2.94) { })()(')()(' 12121 βββββ νννν KIIKdk −−= , (2.95) kp k d r , 1 1 j− =β , (2.96) kp k d r , 2 j =β , (2.97) kr k p , , µ µ ν φ = , (2.98) and [Tk] is called transfer matrix for region k. µφ,k and µr,k are the permeability of the layer k in the φ and r directions respectively. The Bessel functions are of the modified first and second kind, of the order ν. 2.6 Computation procedure developed during the work The practical analysis in this work is based on the MTLM method. The MLTM analysis was programmed to compute the electromagnetic field quantities and power flow at any boundaries between all layers, once By,k or Hx,k is given at any particular boundary. The procedure uses the rectangular multi-layer model of the rotor (Fig. 2.3) and it is commenced by assuming a low
  59. 59. 60 value of tangential field strength Hx,1 at the inner rotor surface and by calculating the corresponding normal component of flux density B1. The MLTM technique then evaluated By,k and Hx,k at all rotor 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.
  60. 60. 61 But also solid rotor induction motors without separate end rings have been designed and tested. For that reason, the study treats the theory which considers the rotor magnetizing flux by dividing it into two components Φ1 and Φ2 and which is originally introduced by Yee (1972).