Electr Eng (2009) 91:35–49 DOI 10.1007/s00202-009-0114-1 ORIGINAL PAPER Electromagnetic and mechanical design aspects of a high-speed solid-rotor induction machine with no separate copper electric circuit in the megawatt range Juha Pyrhönen · Janne Nerg · Aki Mikkola · Jussi Sopanen · Tuomo Aho Received: 5 August 2008 / Accepted: 14 April 2009 / Published online: 30 April 2009 © Springer-Verlag 2009 Abstract The most crucial electro-magnetic and mechanical design aspects of an integrated electrical-motor–gas-compressor system in high speed and high power operation are presented. The electric motor type considered is a solid-rotor induction motor with properties of which are particularly well suited in high-speed operation. The effect of the electro-magnetic material properties of the solid rotor core material on the performance of the machine is discussed. Guidelines to improve the performance of the solid-rotor induction motor are given. Thermal design aspects of a solid-rotor induction motor are presented. The mechanical properties of a solid rotor are discussed. Bearing arrangements as well as the rotor dynamics of an integrated system are presented. Keywords Solid rotor induction motor · Solid rotor materials · Electromagnetic design · Thermal design · Mechanical design · Bearing arrangements · Rotor dynamics Abbreviations FEM Finite element method IGCT Integrated gate commutated thyristor IGBT Insulated gate bipolar transistor SFD Squeeze film damper AMB Active magnetic bearing BW Backward whirling FW Forward whirling 1 Introduction High-speed applications such as large gas compressors, large vacuum systems, large waste-water treatment plant blowers J. Pyrhönen · J. Nerg (B) · A. Mikkola · J. Sopanen · T. Aho South Karelia University of Applied Sciences, Lappeenranta, Finland e-mail: nerg@lut.fi etc. are often operated by traditional, existing technology, where gas turbines are utilized as primary movers to drive the compressors. No gear is needed between the turbine and the compressor, and therefore these constructions are called direct drives. An alternative is an indirect drive, where a speed-increasing gear driven by a traditional electric motor is used instead. However, advances in high-speed electric motor technology and especially in power electronic converters have made a direct electrical drive an interesting approach. Modern commercially available power electronic converters can provide power up to several megawatts at frequencies up to 200–500 Hz. Three-level IGCT converters may reach 200 Hz and multi-level IGBT converters 500 Hz or more. By using this kind of a power source, the rotational speed of an electrical machine can achieve the value of 12,000–30,000 min−1 . It appears to be more straightforward to design a simple high-speed electromechanical energy conversion device than for instance a complicated gas turbine with a number of rotating components, labyrinth seals and the related lubrication problems, thermal stresses, etc. However, mechanical, thermal, and material constraints of the electrical machine restrict the maximum power and the speed of direct electrical drives. An integrated electric-motor–gas-compressor system has a rotating component that converts the electric energy to mechanical energy. In some cases, it is even possible to attach the working machine directly on the same shaft with the electric motor. In such a case, the motor bearings are used as integrated machine bearings. Such a construction eliminates the need for different ancillary systems such as the gear, the lubrication system for the gear, and especially in the case of an integrated gas compressor, the shaft seal. Furthermore, savings are obtained in volume, losses, and costs. It is noteworthy that the overall performance of an integrated electric-motor–gas-compressor system is defined by 123 36 the operation of individual components such as the motor or the compressor as well as the interactions of components. For this reason, an integrated approach for high speeds has a crucial impact on the electric machine design, where all the design problems are strongly interacting with each other. The electromagnetic, mechanical, and thermal design of the whole drive system must be coupled in the preliminary design stage. This is not only a technical problem, but also requires management consideration. The design of an integrated electric-motor–gas-compressor system is a highly demanding task, which requires a designer to possess a wide knowledge of all the engineering disciplines associated with the performance of the system. To find an optimal design, different engineering teams should co-operate with each other. The co-operation is productive if the physical requirements of the electromagnetic, mechanical, and thermal design aspects are understood and taken into account. The objective of this study is to introduce the most crucial design aspects that should be examined when an integrated electric-motor–gas-compressor system is considered. This will be accomplished by explaining the requirements of the electromagnetic and mechanical performance of integrated systems. The electrical machine type addressed in this study is a solid-rotor induction motor with no rotor windings. Such a motor is particularly well suited for a highspeed, high-power application, where mechanical aspects are of major importance. 2 Machine types for high-speed energy conversion In general, the electromagnetic design of an electric motor can be assumed to be independent of the machine speed, while the mechanical properties of the drive system usually dominate the design. There is an increasing interest in using permanent magnet technology in large high-speed applications. The mechanical design aspects are, however, challenging when permanent magnet machines are built for large peripheral speeds [1–3]. An induction machine is traditionally manufactured of laminations and a die-cast or soldered squirrel cage. This kind of a construction is mechanically undetermined as it contains hundreds of individual bodies, which may move with respect to each other. For this reason, such a motor usually has no deterministic balance. Instead, the balancing of the rotor varies with respect to the rotating speed. It is noteworthy that the assembled stacks of laminations increase the weight of the rotor, while their capability to carry bending or shear loading is insignificant. Consequently, a laminated rotor is vulnerable to mechanical vibration, which easily leads to problems at high speeds. The mechanical strength of electrical steels limits the surface speeds of laminated rotors to about 200 m/s. At higher speeds the construction may not be used at all. In order 123 Electr Eng (2009) 91:35–49 to overcome problems associated with the mechanical design of a laminated rotor, a solid rotor with a squirrel cage can be used. Manufacturing of such a rotor is, however, expensive. The solid rotor with a squirrel cage still suffers from the problem that the rotor consists of several individual bodies. As different materials are used in the squirrel cage and the rotor body, loads caused by heat expansion and centrifugal forces require careful considerations on the mechanical design of the rotor. In some cases, the squirrel cage is welded using for instance diffusion welding to the rotor core. This sets high requirements for the quality of the welding process leading to high manufacturing expenses. The pure solid-rotor induction machine with no separate rotor winding is, inherently, the most robust type of electric motors that can be used in high-speed applications. The main aspect from the electromagnetic design point of view is the loss balance in the design. In higher-speed machines, the power density of the machine is considerably higher compared with traditional totally enclosed machines and, furthermore, the losses caused by the cooling gas friction increase in the power of three as the angular velocity of the rotor increases. Hence, the machine designer has to concentrate on motor cooling and minimization of motor losses. Despite the fact that the rotor is made of one solid piece of steel, its mechanical strength and dynamics may become limiting factors in the design. The design of a high-speed motor is a fully coupled engineering problem. 3 Solid-rotor induction motor The solid-rotor induction machine is mechanically robust and resists high angular velocities, but it may not be an excellent motor type from the electromagnetic point of view. The electromagnetic problems related to this motor type are rotor related. When manufacturing the rotor of a single piece of steel, there are no well-conducting windings in the rotor, and therefore, compared for instance with copper-cage rotors, the rotor has to be large enough to produce torque. 3.1 Working principle At synchronous speed, there are no fundamental currents flowing in the rotor of an induction motor, and the flux lines travel directly through the rotor. Some rotor slip frequency is needed to induce voltages and currents in the rotor. The induced rotor axial electric field strength E0 in induction machines is directly proportional to the normal air-gap magnetic field strength H n0 on the rotor surface and the rotor fundamental slip frequency f slip1 as E0 ∼ = f slip1 H n0 . (1) Electr Eng (2009) 91:35–49 37 According to the electromagnetic fundamentals, when a flux is penetrating a solid conductor, the surface impedance Zs adopts in linear and simple cases the form E0 (0, t) E0 (0, t) = Zs = J s (t) n × H n0 (t)
ωslip µ 1+ j = = (1 + j) , δσ 2σ (2) where E0 is the axial electric field at the surface of the solid rotor, n is the unit vector normal to the surface, H n0 is the normal magnetic field strength at the surface of the solid rotor, J s is the surface current in A/m, δ is the depth of penetration, µ is the material permeability, ωslip is the angular frequency of the penetrating field, σ is the material conductivity, and t is time. Consequently, the (1 + j) factor in (2) defines that the active and reactive parts of the surface impedance are equal, and therefore, there is a temporal phase shift of 45◦ between the electric field strength and the surface current. For a non-saturating rotor, this produces a power factor of cos ϕrotor = √1 . 2 Iron, however, saturates and according to Agarwal’s limiting non-linear theory [4], the rotor power factor angle of a fully saturated smooth rotor is 26.6◦ , which gives a rotor power factor of cos ϕrotor = 0.894. In practice, a smooth rotor power factor is somewhat lower. On the other hand, slitting the rotor surface with axial slits improves the rotor power factor. According to 2D FEM calculations, a slitted core produces a power factor angle of about 22◦ . The end regions seem to follow Agarwal’s theory. Together with the air gap phenomena, the whole solid-rotor motor power factor typically lies in the range of cos ϕ = 0.7. According to Maxwell’s tension theory, the magnetic field strength between objects in vacuum creates a tension σ F on the object surfaces, which can be written as σF = 1 2 , µ0 Hn0 2 (3) where µ0 is the permeability of vacuum. The tension term can be divided into its normal σ Fn and tangential σ Ftan components as follows σ Fn 1 2 , = µ0 Hn2 − Htan 2  σ F tan = µ0 Hn Htan .  (4) (5) In (4) and (5), Hn and Htan are the normal and tangential components of the magnetic field strength, respectively. In terms of torque production, the tangential component of the tension σ F tan is of significance. In the case of a rotor, the total torque exerted on the rotor can be obtained by integrating the stress tensor over a cylinder that confines the rotor. The surface current density Js creates tangential field strength in an electric machine. The local tangential tension in an air gap can be expressed as follows σ F tan = µ0 Hn Htan = µ0 Hn Js = Bn Js . (6) The equation expresses a local time-dependent value for the tangential stress when the values of the normal flux density Bn and the surface current Js are known. Assuming sinusoidal distributions, an average stress force F tan can be obtained by integrating the stress σ F tan over the rotor surface S = 2πrl as follows, F tan = σ F tan S = 1 B̂n Jˆs S cos ϕrotor , 2 (7) where B̂n and Jˆs are the peak values of magnetic flux density and surface current, respectively. The resulting average force applies at the distance of the rotor radius r from the shaft and, hence, the torque can be written as Tem = F tan r = B̂n Jˆs πr 2 l cos ϕrotor = B̂n Jˆs V cos ϕrotor , (8) where V is the volume of the rotor. Finally, the torque of the solid rotor can be simplified as follows f rslip1 cos ϕrotor V, (9) Tem = B̂n Jˆs cos ϕrotor V ∼ = c B̂n Z where c is a utilization factor. At low slip values, the rotor resistivity dominates while the impedance Z is mainly determined by the rotor resistance. For this reason, the power factor angle remains in a smooth saturable solid rotor within the range of 27–30◦ . In slitted rotors, the angle may be somewhat lower. Accordingly, the torque increases as the slip frequency increases, and the torque decreases as the rotor surface impedance increases. A solid steel rotor, typically, has a relatively high slip frequency f slip1 because the rotor material has a high resistivity compared with traditional rotor conductive materials, aluminium and copper. The losses of the rotor are directly proportional to the slip of the rotor. According to the induction motor theory, the per-unit slip s of the rotor can be defined as a ratio of the rotor slip frequency to the stator supply frequency f s as follows s= f rslip1 s − r ωs − ωr = = . fs s ωs (10) where s is the synchronous rotation angular speed of the machine corresponding to the supply frequency f s and r is the loaded rotor angular speed. In (10), ωs and ωr are the corresponding electrical angular frequencies. Taking the number of pole pairs p into account, the following relations are valid ωs − ωr . (11) ωs = ps ; ωr = pr ; f rslip1 = 2π If the fundamental air-gap power Pδ1 is the fundamental power flowing across the machine air gap from the stator 123 38 Electr Eng (2009) 91:35–49 Table 1 Induction rotor fundamental efficiency as a function of per unit slip for a p = 1 motor f s /Hz f rslip1 /Hz T, per unit Per unit slip Rotor loss, per unit 50 5 1 0.10 1.11 0.1 100 5 1 0.05 2.22 0.1 150 5 1 0.033 3.33 200 5 1 0.025 4.44 250 5 1 0.02 300 5 1 350 5 400 450 500 Speed/min−1 Output power, per unit η1rotor 2700 1 90 5700 2.11 95 0.1 8700 3.22 96.7 0.1 11700 4.33 97.5 5.55 0.1 14700 5.44 98 0.017 6.67 0.1 17700 6.56 98.3 1 0.014 7.78 0.1 20700 7.67 98.6 5 1 0.0125 8.88 0.1 23700 8.77 98.75 5 1 0.011 9.99 0.1 26700 9.88 98.9 5 1 0.01 11.11 0.1 29700 10.99 99 Pδ,per unit The rotor slip frequency is kept at a constant value of f rslip1 = 5 Hz. It can be seen that when the supply frequency increases, the per unit slip decreases and the rotor fundamental efficiency increases to the rotor, the fundamental frequency caused losses of the rotor Pr,loss1 are directly proportional to the per unit slip of the rotor as follows Pr,loss1 = s Pδ1 . winding factor kw1 , the effective air gap length δef , and the number of turns in series Ns as follows L m = µ0 (12) According to (9), a non-zero slip frequency f rslip1 is needed in the rotor to produce torque. Slip frequency also leads to a per unit slip that defines the rotor fundamental efficiency η1rotor , in which the eddy-current losses within the rotor surface, i.e. the rotor surface losses, the rotor hysteresis losses, and the mechanical and gas friction losses are neglected. The rotor fundamental efficiency can be calculated as 2mτ p ′ l (kw1 Ns )2 . π 2 pδef (14) Since 1 τp ∼ = p (15) the magnetizing inductance taking the stator magnetic circuit parts and the air gap into account is inversely proportional to the square of the pole pair number p. Hence, in order to get acceptable power factors p = 1 should, if possible, be used in solid-rotor high-speed induction machines. 3.2 Rotor materials of a solid-rotor induction machine η1rotor Pδ1 − Pr,loss1 Pδ1 − s Pδ1 = = = 1 − s. Pδ1 Pδ1 (13) An example according to Table 1 clarifies this matter. In the example, the effects of supply frequencies are studied by assuming that the desired rated torque is produced using a rotor fundamental slip frequency of f rslip1 = 5 Hz. Table 1 shows that even though the rotor slip frequency f rslip1 on a solid rotor is large, the rotor fundamental efficiency in a high-speed machine can be high. However, it must be kept in mind that with a fixed rotor slip frequency, the per unit slip decreases remarkably when the stator frequency is increased. As a conclusion, an induction machine is at its best when the machine speed is high. The solid rotor itself produces a low power factor because the field pattern created in the material acts as a magnetic and electric conductor. For this reason, the simplest version of the solid rotor—the smooth solid rotor—may not be used in any high-power applications. The rotating field machine magnetizing inductance L m depends on the phase number m, the pole pitch τ p, the effective core length l ′ , the fundamental 123 To produce high torque, the solid rotor material has to meet two main electromagnetic properties: (1) The rotor conductivity should be as large as possible and (2) the rotor material saturation flux density should be as high as possible. Of these two properties, conductivity plays a more important role in terms of torque production. It is noteworthy that the initial permeability of the material is not significant here. In fact, to obtain a considerable effect on the torque production, the initial permeability of the solid rotor material should be as low as 50 [5]. In this section, as a base reference, a 120 kW two-pole, 170 Hz induction motor equipped with a slitted solid iron rotor is used. The main parameters of the reference motor are presented in Table 2. Its synchronous speed is 10,200 min−1 , rated slip 1%, efficiency 0.93, cos φ = 0.61, and the rated torque 110 Nm. The rated current of the motor at 400 V (line-to-line voltage) supply is 300 A. The number of turns per phase is 16. These values are used as reference values when comparing rotors made of different materials. As the main objective of this chapter is to illustrate the effect of different electromagnetic material properties of the Electr Eng (2009) 91:35–49 39 2.00 Table 2 Main parameters of the reference motor 3 Number of stator slots 48 Stator outer diameter 400 mm Stator bore diameter 200 mm Active stator stack length 280 mm Air-gap length 2.5 mm Rotor length 340 mm Rotor slit depth 40 mm Number of rotor slits 34 Rated voltage (line voltage) 230 V Rated phase current 300 A Stator winding connection Delta Rated frequency 170 Hz Rated output power 120 kW rotor material on the performance characteristics of a solidrotor induction motor, a two-dimensional, non-linear timestepping finite-element analysis was selected; in other words, magnetic saturation, skin effect, and motion of the rotor with respect to the stator are taken into account in the analysis. A single-valued magnetization curve was used to model the stator and the rotor. The circuit equations were applied to model the sinusoidal power supply (i.e. the inverter-caused current ripple was neglected) and to take the effect of the stator end fields into account in the calculations. In the twodimensional finite element calculations, the rotor end effects were taken into account by modifying the rotor equivalent resistivity by the Russel end-factor [6]. Special attention was paid to the quality of the finite element mesh on the outer layers of the solid rotor, where the maximum element size was less than one third of the depth of penetration. All the finite element calculations were performed using Flux-2DTM software package from CEDRAT. Figure 1 indicates the effect of the rotor material conductivity on the torque production when the saturation flux density corresponds to the value of iron, Bsat = 2 T. Figure 2 illustrates the importance of the saturation flux density of the rotor material to the rotor torque production. According to the result of Fig. 2, the saturation flux density of the rotor material above the value of 1.25 T seems to have an insignificant effect on the torque. The rated slip remains at the value of 0.67 in materials with Bsat > 1.25 T. Below this value the rated slip and the rotor losses increase dramatically. As the rotor core is acting as a current- and magnetic-fluxcarrying circuit in a solid rotor, a poor power factor of the motor cannot be avoided. However, as we can see in Fig. 3, a high saturation flux density of the rotor material improves 1.75 Electromagnetic torque pu 1 Number of phases 20 µΩcm 30 µΩcm 40 µΩcm 50 µΩcm 60 µΩcm 80 µΩcm 1.50 1.25 1.00 0.75 0.50 0.25 0.00 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Slip [%] Fig. 1 Electromagnetic torque of the solid-rotor induction machine as a function of slip. The rotor resistivity is used as a parameter. The saturation flux density of all the materials is 2 T and the initial relative permeability was set to a constant value of 2,000. We can see that a high conductivity, i.e. a low resistivity, leads to a low slip and a high rotor fundamental efficiency 2.00 Saturation flux density 1.75 Electromagnetic torque pu Number of pole pairs Resistivity 10 µΩcm 1.50 1.25 2.25 T 2.00 T 1.75 T 1.50 T 1.25 T 1.00 T 0.50 T 1.00 0.75 0.50 0.25 0.00 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Slip [%] Fig. 2 Electromagnetic torque of the solid-rotor induction machine as a function of slip. The saturation magnetic flux density is used as a parameter. The resistivity of all the materials is 30 µcm and the initial relative permeability was set to a constant value of 2,000 the power factor. This phenomenon is emphasized when the motor is operated at a low slip. The most important rotor material property, in addition to the fact that the rotor has to be ferromagnetic, is a high conductivity. The resistivity of pure iron at the temperature of 20◦ C is about 9.8 µ cm, which is about four times as high as that of aluminum. A problem from the viewpoint of mechanical strength is that a highly conductive iron may not contain any compounds that are important with respect to the material strength. Adding almost any compound in steel remarkably increases its resistivity. Fig. 4 illustrates the behaviour of iron conductivity with different alloys. 123 Power factor 40 Electr Eng (2009) 91:35–49 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0.00 Saturation flux density 2.25 T 2.00 T 1.75 T 1.50 T 1.25 T 1.00 T 0.50 T 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Slip [%] Fig. 3 Power factor of a two-pole solid-rotor induction machine as a function of slip. The saturation magnetic flux density is used as a parameter. The resistivity of all the materials is 30 µ cm and the initial relative permeability was set to a constant value of 2,000. Similarly as in electromagnetic torque production, the reasonable saturation flux density value is above 1.25 T In electrical engineering applications, silicon and aluminium are added in electrical sheets to maximize the resistivity of the material. As a result, the eddy current losses in the material are minimized. According to Fig. 4, copper could be added in iron to produce an appropriate solid rotor material. Adding 5–6% of copper lowers the saturation flux density of the material to about 1.7 T, which is adequate for the rotor application as explained above. The strength of the alloy is significantly improved when compared with pure iron. The problem associated with the copper–iron alloy is that copper is considered an impurity of steel, and therefore only a few, if any, manufacturers are interested in producing copper iron. The compound of copper requires vacuum casting, which is not widely available. For these reasons, the rotor has to be manufactured using a commercial steel with a low amount of impurities and compounding additives. It is noteworthy that the traditional Fe52 (S355J0/EN 10025) has a room-temperature resistivity of about 25 µcm. 3.3 Improving the performance of solid-rotor induction machine ρ /µΩ cm 80 70 Si 60 Al 50 40 30 Ni 20 Co 10 0 Cu 0 1 2 3 4 5 6 7 8 compounding agent, % per weight Fig. 4 Effect of silicon, aluminium, nickel, cobalt, and copper alloying on the resistivity of iron [7] Fig. 5 Flux lines and flux density distribution of a 170 Hz smooth solid-rotor induction motor at a slip of 1.0%. We can see that the flux penetrates only the rotor surface even though the per unit slip is low 123 The performance of the simplest rotor, viz. a smooth solidsteel rotor, can be analyzed with the finite element approach. Fig. 5 shows the flux pattern in a smooth solid-iron rotor. The motor parameters are the same as presented in Table 2 except that the rotor is smooth. As it can be seen in the figure, the flux does not penetrate very deep into the smooth rotor. The current density is also concentrated on the surface of the rotor while the resistance of the rotor is large. Such a rotor requires a large slip to produce torque thereby resulting in a low efficiency. To improve the performance of the solid rotor, flux penetration into the solid rotor material should be improved. A method to improve the flux penetration into the rotor is to slit the rotor surface as shown in Fig. 6. The parameters of the motor are presented in Table 2. At this point, the coupling of the electromagnetic design with mechanical design becomes obvious, since slitting weakens the mechanical strength of the Electr Eng (2009) 91:35–49 41 Fig. 6 Flux lines and flux density distribution of a 170 Hz slitted solid-rotor induction motor at a per unit slip of 1.0%. Slitting of the solid rotor surface remarkably improves the flux penetration into the rotor Fig. 7 Electromagnetic torque of a 170 Hz solid-rotor induction motor as a function of per unit slip. The rotor slitting depth, presented as a ratio of the rotor radius, is used as a parameter. The best torque is reached when the slitting depth is 60% of the rotor radius. The rated per unit slip of the motor with 60% deep slits is 0.8%. The rotor frequency is hence 1.36 Hz. The smooth solid rotor produces only 38% of the rated torque at this slip rotor as it will be explained in Sect. 5. Due to the mechanical aspects, the maximum slit depth, in practice, is approximately one half of the radius of the solid rotor [8]. Figure 7 shows the considerable effect of slitting on the torque production of a solid rotor construction. 3.4 Additional losses in solid-rotor induction machines The solid rotor surface is sensitive to the air gap harmonics, which may generate a remarkable amount of additional losses on the rotor surface. The distributed winding system of the stator, stator-slot-caused permeance harmonics, and possible time harmonics caused by an inverter supply are sources of spatial harmonics. The designer has, however, several methods that may be used in minimizing the amount of rotor surface losses. The spatial harmonics may be minimized by using stator winding Fig. 8 Two-pole 9 MW, 200 Hz, solid rotor losses at the rated slip of s = 1.2% with different resistivities of the rotor coating material. We can see that the coating with a resistivity of about two times that of the core material or more reduces the total rotor losses arrangements that produce a minimum amount of harmonics. For example, chording of the stator winding effectively reduces the lower space harmonics in the air-gap flux. The permeance harmonics may be minimized by intelligent magnetic circuit design. The air gap time harmonics may be minimized for instance by using high switching frequency inverters and filtering. From the rotor material point of view, there are two alternatives to minimize the rotor surface additional losses caused by the harmonics. It is either possible to use a well-conducting non-magnetic coating or a high-permeability low-conducting material. The material allows the harmonics attenuate before entering the rotor core and acts like a mirror while it does not let the harmonics penetrate the rotor core, where additional losses are likely to occur. Figure 8 indicates the results for the rotor coating alternatives when a 9 MW 12,000 min−1 rotor is coated with different materials having a saturation flux density of 1 T and a relative permeability of 50. The main parameters of the 9 MW solid-rotor induction motor are presented in Table 3. 123 42 Electr Eng (2009) 91:35–49 Table 3 Main parameters of 9 MW solid-rotor induction motor with a slitted solid iron rotor Number of poles 2 Number of phases 3 Rated output power 9000 kW Rated frequency 200 Hz Stator outer diameter 750 mm Stator bore diameter 335 mm Active stator length 850 mm Rotor outer diameter 325 mm Number of stator slots 48 Number of rotor slits 28 Width of a rotor slit 4 mm Depth of a rotor slit 60 mm Figure 9 shows the coating principle. The coating may be manufactured for instance by using a stainless ferromagnetic tube, which may be welded to every tooth and the ends. This procedure also significantly improves the mechanical strength of the rotor. 4 Thermal design of a solid-rotor induction machine In electrical machines, the design of heat transfer is of equal importance as the electromagnetic design of the machine, since the thermal rise of the machine eventually decides the output power of the machine. When high-speed machines are considered, the thermal aspects are emphasized. This is due to the high power density and the increased friction losses. Usually, high-speed machines are cooled by forced convection using open circuit air-cooling. The cooling gas friction loss Pfr in the air gap can be written as [9] Pfr = k1 C T ρπ ω3r 4 l, Fig. 9 Coating principle studied in Fig. 8. The core material is slitted, and on the surface of the rotor, there is a uniform 3–5 mm thick ferromagnetic coating. The coating smoothens the rotor surface. A smooth surface also reduces the friction losses. A highly resistive ferromagnetic coating also reduces the total rotor losses 123 (16) where C T is the torque coefficient, ρ is the mass density of the fluid, ω is the angular velocity of the rotor, r is the rotor radius, l is the length of the rotor, and k1 is the roughness coefficient, the value of which is 1.0 for a smooth rotor and typically 2–4 for axially slitted rotors. The effect of the rotor slitting is twofold: the friction losses increase, whereas the slitting intensifies the cooling of the rotor because of the increased turbulence level and heat transfer surface. In axial cooling, the gas flows through the air-gap, which is generally the case in high-speed machines, and thus the friction losses increase. The rotor forces the cooling gas into a tangential movement and, accordingly, some power is needed to accelerate the cooling gas. If the radial air-gap length is small compared with the rotor radius, the axial flow power loss Pfr,a can be approximated as follows Pfr,a = k2 qm u 2 , (17) where k2 is the velocity factor, qm is the mass flow rate of the cooling gas, and u is the peripheral speed of the rotor. It is important to note that the gas friction loss Pfr turns into heat in the air gap, but the mass flow rate dependent loss component Pfr,a turns into heat mostly after the air gap, i.e. in the end-winding space. As shown in Eq. (16), the friction loss is proportional to the third power of the angular velocity of the rotor and to the fourth power of the rotor radius. The mass flow rate dependent loss component is proportional to the square of the peripheral speed of the rotor and directly proportional to the mass flow rate of the cooling fluid, respectively. The desired angular velocity, peripheral speed, and the rotor radius are results from the electromagnetic and mechanical analyses. Thus, the only parameter which can be utilized in decreasing the axial temperature gradient of the machine, and subsequently in minimizing the power of the auxiliary blower, is the axial length of the cooling gas flow path. The most common solution is the utilization of radial cooling ducts. This can be accomplished by dividing the stator stack into two parts, and the cooling fluid is blown inside Coating Slitted solid-rotor Slitted solid-rotor Electr Eng (2009) 91:35–49 43 Fig. 10 Cooling channels of a solid-rotor induction motor with a slitted solid rotor Table 4 Main parameters of 430 kW solid-rotor induction motor with a slitted solid iron rotor Number of poles 2 Number of phases 3 Rated output power 430 kW Rated frequency 170 Hz Stator outer diameter 391 mm Stator bore diameter 242 mm Active stator length 310 mm Rotor outer diameter 237 mm Number of stator slots 60 Number of rotor slits 40 Width of a rotor slit 2.5 mm Depth of a rotor slit 50 mm the machine through a radial channel. The cooling fluid flow is divided into two equal parts in the air-gap area, both eliminating half of the rotor and friction losses of the machine. A typical cooling system of a high-speed machine is illustrated in Fig. 10. The utilization of lumped parameters, i.e. thermal resistance networks, has proven to be an accurate enough calculation tool for the thermal analysis of solid-rotor induction motors [9,10]. In the complete thermal resistance network model of a solid-rotor induction motor, both the axial and radial heat transfer inside the machine, the presence of the contact transition layers, and the heating of the cooling air are taken into account. As an example of the lumped-parameterbased thermal analysis of a solid-rotor induction motor, let us consider a case where the end-winding temperature is minimized by blowing a part of the cooling air directly to the endwindings through additional bores in the frame. The analyzed machine was a three-phase two-pole 430 kW 170 Hz solidrotor induction motor with a slitted solid rotor. The stator stack was divided into two sections, between which there is a radial cooling duct. The rotor was equipped with copper end rings. The main parameters of the analyzed 430 kW solid-rotor induction motor are shown in Table 4. Fig. 11 Effect of the additional end-winding cooling on the temperatures of a 450 kW, high-speed induction machine. The end-winding temperature can be effectively decreased by blowing a part of the total volume rate of the cooling fluid directly to the end-windings, The results are calculated applying a lumped-parameter-based thermal analysis. The incoming fluid temperature is 35◦ C The temperature distribution of the motor analyzed was calculated at the nominal operation point, i.e. the torque was 405 Nm and the rotational speed of the rotor was 10,090 min−1 . The calculated temperatures of the end-windings, the coils in the slots, the rotor teeth, and the rotor end using two different cooling fluid flow paths are illustrated in Fig. 11. The volume rate of the cooling air entering the machine was 0.367 m3 /s in both cases. The first, left-hand bars in the figure correspond to the case in which all the cooling fluid was flowing through the radial cooling duct to the air-gap, and the second, right-hand bars correspond to the case in which 30% of the total volume rate of the cooling fluid was blown directly to the end-windings; in other words, 70% of the total volume rate of the cooling fluid was flowing through the radial cooling duct. Because there are two cooling ducts for the end-windings, the volume rate of the cooling air blown directly to the drive and non-drive end end-windings was 0.055 m3 /s. The results are calculated with a lumpedparameter-based thermal analysis in steady-state conditions. The thermal resistance network used consists of 15 nodes. The details of the thermal model are reported by the authors in [10]. The calculation results, in which a part of the cooling air was blown directly to the end windings, were verified by measuring the average value of the end-winding temperature, the temperature of the coils in the slots with Pt100 sensors. The temperature of the inlet air as well as the volumetric air flows through the radial cooling duct and the end-winding bores were measured. The temperatures of the rotor end ring and at the bottom of one rotor teeth were measured with a thermometer after the machine had stopped. A comparison 123 44 Electr Eng (2009) 91:35–49 Table 5 Comparison of the calculated and measured temperatures of the 430 kW solid-rotor induction motor in degrees celsius Location Measured Calculated Coils in slots 87 88 End-winding 122 123 Rotor teeth 88 90 Rotor end ring 90 92 of the calculated and measured temperatures in the case of additional end-winding cooling is presented in Table 5. 5 Mechanical design of a solid-rotor motor In the mechanical design of a solid-rotor motor, a number of static and dynamics issues should be considered. In terms of dynamics, a solid rotor provides considerable mechanical advantages compared with other mechanical constructions of the rotor. The mechanical advantages of a solid-rotor construction are based on the facts that the rotor does not contain welded, friction, or compression joints and it consists of only one part made of a single piece of material. 5.1 Properties of solid rotor As solid rotors are machined from one piece of steel bar, the structure is straightforward to balance out. It is noteworthy that in the case of a solid rotor, balancing does not have to be reproduced during the lifetime of the solid rotor, as may be the case with alternative structures. The solid rotor does not contain many mechanical parts such as steel laminations, which makes it difficult to evaluate the stiffness and damping properties of the rotor. The capability of steel laminations to carry shear and bending loading is usually low, and it depends on the assembly procedure. For example, Garvey et al. [11] concluded that for laminated stacks, the Young’s modulus values for axial extension-compression are in the order of 0.8×109 N/m2 , and the shear modulus values are in the order of 0.3×109 N/m2 . In addition, a wide range of effective moduli can be obtained with different clamping pressures and surface treatments of individual laminations. As a result, the mechanical properties of apparently similar laminated rotors may vary. This, in turn, forces designers to use large safety factors in order to avoid dynamic problems resulting from operation close to natural frequencies of the rotor. Solid rotors equipped with a squirrel cage are mechanically robust compared with laminated squirrel cage rotors. However, in squirrel cage solid rotors, two materials, viz. steel and copper, are jointed together. These mechanical joints are loaded by centrifugal inertial forces resulting from different material densities and thermal loading caused 123 Fig. 12 Finite element model used in determining the effect of slitting to the torsion stiffness of the solid rotor. The outer diameter of the rotor is 313 mm and the diameter of the non-slitted cross-section is 193 mm by different coefficients of thermal expansion of steel and copper. For this reason, a high-quality joining process is needed when squirrel cage solid rotors are manufactured. Although purely solid rotors do not suffer from mechanical disadvantages similarly as laminated or squirrel gage solid rotors, the slitting of the rotor in order to improve the flux penetration will decrease the mechanical properties of the rotor. In practice, the rotor design is a trade-off between the electromagnetic and the mechanical properties of the rotor.The mechanical drawbacks of slitting are related to the decreased torsion stiffness and increased vulnerability to fatigue damage. The slitting of the rotor produces geometrical discontinuities, i.e. notches in which a significant stress concentration can occur. As a result, notches can be places of fatigue crack initiation, and subsequent crack propagation can cause severe damage to the structural component. The largest cyclic stresses in a high-speed rotor are caused by changes in the rotational speed. The major stress variation is caused during a run-up from standstill to the maximum rotation speed of the motor. In addition, variable-speed motors may experience smaller stress cycles when the rotation speed is controlled to achieve an optimal process performance [8]. The torsional stiffness of the slitted rotor is considerably lower compared with a smooth solid rotor. The polar moment of inertia J that describes the ability to resist torsion can be calculated for solid circular cross sections as follows π Dr4 , (18) 32 where Dr is the diameter of the rotor. The torsion stiffness of a slitted rotor can be studied using a detailed finite element model, as shown in Fig. 12. In the rotor depicted in Fig. 12, the effective diameter in terms of polar moment of inertia is J= Electr Eng (2009) 91:35–49 about 108% of the diameter of the non-slitted cross section of the rotor. In other words, the torsion stiffness of the slitted rotor in Fig. 12 is 19.7% of the torsion stiffness of an equal-sized non-slitted solid rotor. In practice, these issues suggest that the designer should use the diameter of the nonslitted cross-section of the rotor when evaluating the torsion stiffness of the solid rotor. Fastening a thin smooth ferromagnetic layer on the slitted rotor surface restores the torsional stiffness of the rotor. Analytical methods for determining the torsion stiffness of a slitted rotor are not straightforward. In practice, the designer should use the finite element method for reliable torsional stiffness evaluation. However, the effect of coating on the torsion stiffness of the rotor can be estimated analytically. The polar moment of inertia of coating, i.e. a thin-walled tube, can be evaluated as follows   4 4 − Din π Dout , (19) Jp = 32 where Dout is the outer diameter and din is the inner diameter of the tube, respectively. If the rotor in Fig. 12 is coated with a 3 mm thick steel plate in such a way that Dout = 313 mm and din = 307 mm, the torsion stiffness of the rotor increases by 62% compared with a non-coated rotor. This is because the polar moments of inertia of the slitted rotor and the coat can be added together. It is noteworthy that a detailed finite element calculation indicates even a 100% increase in the torsion stiffness when the coating is carefully welded to the rotor. Accordingly, the fastening method of the coating has a significant effect on the torsional stiffness. For example, if the coating is fastened by laser welding, in which the weld is narrow, the stiffening effect of the coating is significantly lower than in the case of a fully welded coating. Therefore, the designer should use analytically obtained values from (18) and (19) since the obtained torsion stiffness values are lower than in reality, i.e. the results are always on the safe side. 5.2 Bearing arrangements The selection of the bearing arrangement of a solid-rotor motor can be made in various ways. As solid rotors are connected to impellers without a gear, the bearing arrangement must support the weight of the rotor and the impellers. The impellers can be connected to the rotor using a rigid shaft hub coupling such as a conical sleeve or tapered press. In most practical cases, machines contain two impellers, one at both ends of the shaft. However, it is possible to make a compressor with one impeller only. This will affect the bearing arrangement design, since the impeller causes an axial force that is directed outwards. This axial force is almost compensated in two-sided impeller applications, whereas in 45 one-sided impeller applications the force must be compensated by the bearings of the motor. The suitable bearing type depends on the size and the operating conditions of the motor. Usually, in motors with a rated power below 2 MW and operating speeds below 15,000 min−1 the rotors can be supported with oil-lubricated ball or spindle bearings. Tilting pad journal bearings can be an appropriate choice when the weight of the rotor becomes considerable. The drawback of the journal bearings is their larger frictional losses compared with rolling element bearings. However, in larger units, the rated power of which is above 2 MW, the percentile losses become insignificant. A benefit of journal bearings is their larger damping capability, which makes it possible to design a motor that operates above its first critical speed. A possible alternative for the bearings of a solid-rotor motor could be spindle bearings equipped with squeeze film dampers (SFD). In SFD applications, the outer ring of the rolling element bearing can move radially in a cap filled with oil. The oil film between the nonrotating surfaces provides damping in the radial direction without large frictional losses of journal bearings. The area of active magnetic bearings (AMBs) has recently been intensively developed because this non-contact support system has numerous advantages compared with conventional bearings. The most important advantages are almost non-existent friction and consequently, a low energy loss, no need for lubrication, quiet operation, and adjustable stiffness and damping, which makes accurate rotor positioning possible. In addition, AMBs offer almost unlimited control over the rotor that they support. Adjustable stiffness and damping are helpful especially from the mechanical point of view. Adjustable stiffness of the AMB makes it possible to decrease the natural frequencies of the rigid body modes of the rotor under the operation speed, while the damping coefficient can be increased. As a consequence, the vibration at the natural frequency of the rigid body damps out, and the motor can be used at supercritical speeds. Furthermore, unbalance compensation during motor operation is possible because of the active feedback control of AMB. In the unbalance compensation, the rotor does not rotate around its geometrical centre but rotates around the centre of mass. It is obvious that this is not possible when using mechanical bearings. Typically, an AMB application is expensive and unique owing to the high expense associated with the development of control software. However, because of their numerous advantages, AMBs are an appropriate choice for large-scale solid-rotor motors, the rated power of which is several megawatts. Further, improved materials, strategies of the controller, and electric components are enhancing the performance and reliability of AMB. Despite this, additional bearings, i.e. retainer bearings, have a vital role in the AMB applications. 123 46 Electr Eng (2009) 91:35–49 Fig. 13 Schematic diagram of a rotor with two impellers. The finite element node numbers are shown above the centre line. The slitted part of the rotor is between nodes 10 and 13. The dimensions are in millimetres 5.3 Rotor dynamics of solid-rotor motor Rotor dynamic analysis constitutes a crucial part in the design of a high-speed solid-rotor motor. Natural frequencies of the rotor system as well as vibration responses caused by excitation forces have to be determined. It is important to point out that both experimental and analytical methods should be utilized in the rotor dynamic analysis. In general, the following analysis should be performed for a solid-rotor motor. First, the free-free vibration modes, i.e. unsupported modes should be determined for the rotor-impeller assembly. This can be accomplished using the finite element method. The obtained results should be verified with experimental modal analysis. Uncertain model parameters such as the stiffness of the impeller attachment to the rotor can be determined using experimentally obtained results. The second step in the rotor dynamic analysis is the evaluation of critical speeds of the rotor bearing system. It is important to note that because of the gyroscopic effects and speed-dependent dynamic coefficients of the bearings, the natural frequencies of the rotating shaft depend on the angular velocity. In general, the natural frequencies of the rotor bearing system can be divided into backward whirling modes (BW) and forward whirling (FW) modes. In the case of the BW mode, the rotor whirling direction is opposite to the rotor rotation direction, while in the FW mode the whirling direction is the same as the rotor rotation direction. Usually the natural frequency of the BW mode decreases as the rotation speed increases. Correspondingly, the natural frequency of the FW mode increases as the rotation speed increases. A critical speed occurs when a natural frequency of the rotor bearing system coincides with the rotating speed of the rotor. A Campbell diagram, where the natural frequencies of the rotor-bearing system are shown as a function of rotation speed, is a useful tool in finding the critical speeds of the rotor bearing system. Usually, critical speeds are excited by unbalance forces that affect the rotor. It is important to note that not all vibration modes are excited by unbalance. 123 For example, if bearing stiffnesses are symmetrical in two radial directions, unbalance forces cannot excite the backward whirling modes. Normally, a third step in the analysis is the calculation of the steady state responses caused by exciting forces, such as rotor residual unbalance. In this step, the rotor dynamic designer should use his or hers judgment to ensure that the vibration responses are in an acceptable level in the whole operating speed range. If any problems are found, necessary modifications to the design should be made and the analysis should be repeated until a satisfactory solution is found. As an example of the rotor dynamic analysis of a solidrotor motor, a structure shown in Fig. 13 is studied. The solidrotor with two impellers is modelled using shear deformable beam finite elements. Only lateral degrees-of-freedom of the rotor are taken into account in the analysis, i.e. longitudinal and torsion vibrations are not studied. The analysis is accomplished in MatlabTM software [12] employing the general rotor dynamic theory presented in [13–15]. The rotor is supported by two ball bearings that have different stiffness and damping coefficients in two radial directions (y and z). The impellers are modelled as rigid disks that are attached to the shaft using translational and rotational springs. In addition, the centre of gravity of both impellers is located at a distance of 65 mm from the impeller attachment to the shaft. The rotor is horizontally mounted, and gravity affects in the negative z-direction (g = 9.81 m/s2 ). The total mass of the rotor with impellers is 155.2 kg. The parameters used in the modelling are shown in Table 6. The Campbell diagram of the studied rotor is shown in Fig. 14. Four critical speeds can be seen at the points where the rotation speed and the natural frequencies intersect. Figure 15 shows the steady-state unbalance responses of the rotor-bearing system. We can see that the critical speed of BW mode at 8,350 min−1 is slightly excited. The reason for this is that the bearing stiffnesses are asymmetrical. In reality this phenomenon can take a place if the ball bearings are not properly preloaded. However, this first critical speed may Electr Eng (2009) 91:35–49 47 Table 6 Parameters used in the rotor dynamic analysis of the solid-rotor motor shown in Fig. 13 Young’s modulus of the rotor 210,000 MPa Material density of the rotor 7,800 kg/m3 Poisson’s ratio of the rotor 0.3 Properties of the slitted part of the rotor Area 0.0251 m2 Second moment of area, I yy , Izz 6.2832 × 10−5 m4 Bearing stiffness coefficients Horizontal kby Vertical kbz Bearing damping coefficients 1.9 × 108 N/m 2.1 × 108 N/m Horizontal cby 4,750 Ns/m Vertical cbz 5,250 Ns/m Impeller mass properties Mass 19.0 kg Polar mass moment of inertia 0.5 kg m2 Diametral mass moment of inertia 0.3 kg m2 Stiffness of impeller attachment Translational Rotational Rotor unbalance masses 1.0 × 1011 N/m not cause problems because the calculated response is small, and even slight damping will diminish it. The first dangerous critical speed is that of the FW mode at 12,950 min−1 . If a 20% safety margin is adopted, the rotor could be operating up to a rotation speed of 10,300 min−1 . It can also be seen that the third BW mode at 13,500 min−1 is excited because of the asymmetrical support of the rotor. Figure 16 shows the whirling modes of the rotor at critical speeds. The analysis example above clarified the phases in the rotor dynamic analysis that should be carried out in the design phase of a new solid-rotor motor. It is noteworthy, however, that a computer simulation should not be used as the only tool in the design process. Vibration measurements are an essential part of the product development, and they should be used in parallel with the modelling approach. Physical rotors may contain several parameters that cannot be evaluated without experimental measurements. For example, the determination of the parameters of the rotor’s supporting structure usually requires measurements. The rotor dynamics model can then be updated and tuned with the help of experimentally obtained results. 1.5 × 106 Nm/rad D1 Impeller (node 23) 36 g mm @ 270◦ D1 Rotor end (node 9) 50 g mm @ 0◦ D2 Rotor end (node 14) 50 g mm @ 0◦ D2 Impeller (node 24) 36 g mm @ 90◦ 6 Conclusions Large impellers in the applications of gas compressors, vacuum systems, and waste-water treatment systems can be Fig. 14 Campbell diagram of the studied solid-rotor motor 123 48 Electr Eng (2009) 91:35–49 Fig. 15 Steady-state unbalance response of the studied solid-rotor motor with two impellers Fig. 16 Rotor whirling modes that are excited at critical speeds shown in Fig. 15. a–d Show the whirling motion of the rotor centre line. a Second BW mode at 8,350 min−1 , f = 139.1 Hz, ξ = 0.16%, b firs FW mode at 12,950 min−1 , f = 215.8 Hz, ξ = 1.4%, c third BW mode at 13,500 min−1 , f = 225.1 Hz, ξ = 1.4%, d second FW mode at 16,650 min−1 , f = 277.2 Hz, ξ = 1.4%, where f is frequency and ξ damping factor of the mode operated by a solid-rotor motor. In these applications, the impeller can be attached to the solid-rotor motor directly without a gearbox or a flexible coupling. This paper explains the main design principles behind the electrical and mechanical design of a solid-rotor motor. The design of a solid-rotor motor is a multidisciplinary problem, in which the electrical, thermal, and mechanical design aspects should simultaneously be taken into account. In practice, this can be accomplished by extensively using computer simulations. Computer simulations can serve as 123 a communication media within a multidisciplinary product development team. References 1. Arkkio A, Jokinen T, Lantto E (2005) Induction and permanentmagnet synchronous machines for high-speed applications. In: Proceedings of the eighth international conference on electrical machines and systems, vol 2, Nanjing, China, pp 871–876 2. Bumby JR, Spooner E, Dellora G, Gstrein W, Sutter H, Tennant H, Wagner J (2004) Electrical machines for use in electronically Electr Eng (2009) 91:35–49 3. 4. 5. 6. 7. 8. assisted turbo-chargers. In: Proceedings of IEE conference on power electronics, machines and drives, Edinburgh, UK, pp 344– 349 Binder A, Schneider T, Klohr M (2006) Fixation of buried and surface-mounted magnets in high-speed permanent-magnet synchronous machines. IEEE Trans Ind Appl 42(4):1031–1037 Agarwal PD (1959) Eddy-current losses in solid and laminated iron. Proc AIEE 42(1):169–181 Aho T, Sihvo V, Nerg J, Pyrhönen J (2007) Rotor materials for medium-speed solid-rotor induction motors. In: Proceedings of IEEE conference on electric machines and drives IEMDC, vol 1, Antalaya, Turkey, pp 525–530 Russel R, Norsworthy K (1958) Eddy current and wall losses in screened induction motors. Proc IEE 105(20):163–173 Heck C (1974) Magnetic materials and their applications. Butterworth, London Aho T, Nerg J, Sopanen J, Huppunen J, Pyrhönen J (2006) Analyzing the effect of the rotor slit depth on the electric and mechanical performance of a solid-rotor induction motor. Int Rev Electr Eng (IREE) 1(4):516–524 49 9. Saari J (1998) Thermal analysis of high-speed induction machines, Acta Polytechnica Scandinavica. Electrical Engineering Series no. 90, Diss. HUT, Espoo, Finland 10. Nerg J, Rilla M, Pyrhönen J (2008) Thermal analysis of radial-flux electrical machines with a high power density. In: IEEE Trans. Ind Electron 55(10):3543–3554 11. Garvey SD, Penny JET, Friswell MI, Lees AW (2004) The stiffening effect of laminated rotor cores on flexible-rotor electrical machines. In: Proceedings of the eighth international conference on vibrations in rotating machinery, Swansea, UK, pp 193–202 12. MATLAB Help (2003) Program manual (electric), version 6.5, Mathworks Inc 13. Chen WJ, Gunter EJ (2005) Introduction to dynamics of rotor-bearing systems. Trafford Publishing, Victoria 14. Krämer E (1993) Dynamics of rotors and foundation. Spinger, Berlin 15. Lalanne M, Ferraris G (1998) Rotordynamics prediction in engineering, 2nd edn. Wiley, West Sussex 123