A comprehensive overview of high-speed solid-rotor induction machines: Applications, classification, and multi-physics modeling

Contents lists available at ScienceDirect International Journal of Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes Review A comprehensive overview of high-speed solid-rotor induction machines: Applications, classification, and multi-physics modeling Vladimir Bileka,∗, Jan Bartaa, Marek Tomana, Petr Losakb, Gerd Bramerdorferc aDepartment of Power Electrical and Electronic Engineering, Brno University of Technology, Brno, 616 00, Czech Republic bInstitute of Solid Mechanics, Mechatronics and Biomechanics, Brno University of Technology, Brno, 616 69, Czech Republic cInstitute of Electrical Drives and Power Electronics, Johannes Kepler University, Linz, 4040, Austria A R T I C L E I N F O Keywords: Finite element method High-speed electrical machines Induction machines Multi-physical optimization Solid-rotor machines A B S T R A C T Solid-rotor induction machines have gained significant attention in various industrial applications due to their robustness, reliability, and cost-effectiveness. This paper presents a comprehensive overview of these machines, covering their classification and various applications. The paper starts with discussing the widespread usage of solid-rotor induction machines in numerous industry sectors, including manufacturing, transportation, and renewable energy generation. The ability to operate under harsh environmental conditions and in safety-critical settings has made these machines indispensable in many fields of engineering. Their detailed classification based on different rotor topologies is provided, highlighting the unique design features and performance characteristics of each category. Simple and hybrid configurations and their distinct advantages and limitations in specific applications are included. This paper further explores the essential aspects of multi-physics modeling of solid-rotor induction machines, incorporating electromagnetic, mechanical, and thermal considerations to gain deep insights into the complex interactions between components and to guide the optimization process for enhanced performance and efficiency. This work is intended as a valuable reference for researchers and engineers seeking a comprehensive understanding of solid-rotor induction machines, from their diverse applications to the intricacies of their electromagnetic, thermal, and mechanical modeling. By shedding light on these aspects, this work contributes to the advancement and utilization uptake of these machines in modern industrial settings. Contents 1. Introduction ...................................................................................................................................................................................................... 2 2. Overview of solid-rotor IM applications ............................................................................................................................................................... 3 2.1. Gas compressor and turbine applications.................................................................................................................................................. 3 2.2. Flywheel energy storage systems ............................................................................................................................................................. 4 2.3. High-speed spindle applications............................................................................................................................................................... 4 2.4. Turbomolecular pumps ........................................................................................................................................................................... 4 3. Overview of solid-rotor IM topologies.................................................................................................................................................................. 5 3.1. Smooth solid-rotor IM............................................................................................................................................................................. 5 3.2. Axially slitted solid-rotor IM.................................................................................................................................................................... 5 3.3. Solid-rotor IM with radial rotor surface grooves........................................................................................................................................ 7 3.4. Squirrel-cage solid-rotor IM..................................................................................................................................................................... 7 4. Suitable materials for solid rotors........................................................................................................................................................................ 8 5. Electromagnetic modeling of solid-rotor IMs ........................................................................................................................................................ 9 5.1. Analytical methods................................................................................................................................................................................. 9 5.2. Numerical methods................................................................................................................................................................................. 9 5.2.1. 2D modeling and numerical electromagnetic calculations of induction machines ........................................................................... 10 5.2.2. Corrective end-effect factors for a solid-rotor.............................................................................................................................. 11 ∗Corresponding author. E-mail addresses: [email protected] (V. Bilek), [email protected] (J. Barta), [email protected] (M. Toman), [email protected] (P. Losak), [email protected] (G. Bramerdorfer). https://doi.org/10.1016/j.ijepes.2025.110520 Received 8 May 2024; Received in revised form 21 December 2024; Accepted 4 February 2025 Electrical Power and Energy Systems 166 (2025) 110520 Available online 19 February 2025 0142-0615/© 2025 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ). V. Bilek et al. 5.2.3. Corrective end-effect factors for a smooth solid-rotor with copper coating .................................................................................... 13 5.2.4. Corrective end-effect factors for a solid-rotor with highly conductive end rings ............................................................................. 13 5.2.5. Corrective end-effect factors for a solid-rotor with radial rotor surface grooves ............................................................................. 13 6. Cooling and thermal modeling of solid-rotor induction machines........................................................................................................................... 14 6.1. Cooling concepts for high-speed machines................................................................................................................................................ 15 6.2. Overview of commonly used thermal modeling techniques ........................................................................................................................ 15 6.2.1. Thermal models utilizing computational fluid dynamics .............................................................................................................. 15 6.2.2. Thermal models utilizing a lumped parameter thermal network ................................................................................................... 15 6.2.3. Thermal models utilizing finite element method ......................................................................................................................... 16 6.3. Thermal modeling using a lumped parameter thermal network .................................................................................................................. 16 6.3.1. Conductive heat transfer........................................................................................................................................................... 16 6.3.2. Convective heat transfer ........................................................................................................................................................... 17 6.3.3. Radiation heat transfer ............................................................................................................................................................. 18 6.3.4. Modeling of coolant flow .......................................................................................................................................................... 18 7. Mechanical modeling of solid-rotor induction machines ........................................................................................................................................ 19 7.1. Stress calculation.................................................................................................................................................................................... 19 7.1.1. Load from rotation ................................................................................................................................................................... 19 7.1.2. Temperature load..................................................................................................................................................................... 20 7.1.3. Unbalance ............................................................................................................................................................................... 20 7.1.4. Unbalanced magnetic pull (UMP) .............................................................................................................................................. 21 7.1.5. External loads.......................................................................................................................................................................... 21 7.2. Critical speed calculation ........................................................................................................................................................................ 21 8. General overview of rotor topology characteristics ............................................................................................................................................... 22 9. Conclusion ........................................................................................................................................................................................................ 23 CRediT authorship contribution statement ........................................................................................................................................................... 24 Declaration of competing interest........................................................................................................................................................................ 24 Acknowledgments .............................................................................................................................................................................................. 24 Data availability ................................................................................................................................................................................................ 24 References......................................................................................................................................................................................................... 24 1. Introduction High-speed electrical machines are becoming increasingly popular in industrial applications, which is reflected in a growing number of publications in recent decades. High-speed, typically means a circumferential speed, a characteristic parameter for rotor strength, is higher than 100 m/s [1]. To achieve high rotation speed, one option is to use a gearbox combined with a standard electrical machine [2]; another option is the utilization of a high-speed electrical machine that enables a direct connection between the machine and the load, such as an impeller, which increases overall system efficiency and reliability [3] compared to a system with a gearbox. Other benefits of high-speed over conventional machines include smaller size and reduced material consumption [4,5]. These are the main factors for increasing industry interest for high-speed electrical machines for numerous applications, such as compressors, vacuum pumps, microturbines, and machining tools [3]. It has been shown that synchronous permanent magnet machines [6, 7] are typically used for higher speeds and lower powers, while, induction machines (IMs) are used for lower speeds and higher powers [3]. In terms of operating performance, it can generally be stated that synchronous machines offer higher power densities, higher power factor, and better efficiency than comparable IMs. Induction machines, however also have advantages, especially in terms of resistance to high temperature and simpler control, and in conjunction with a solidrotor without squirrel-cage, they have the potential to tolerate high mechanical stresses at low manufacturing costs [8]. The solid-rotor is a non-laminated rotor type for IMs that offers high resistance to centrifugal forces, temperature, and environmental influences, which may include, for instance, various harsh chemicals substances. Solid-rotors thus offer high reliability and durability for electrical machines. Their performance can be improved in various ways, for example, by slitting or by adding a copper layer to the rotor surface or the rotor ends. In recent years, various modifications have been discussed and studied in a wide range scientific publications, e.g., [9–14]. These, however focused mainly on the description of one or a few particular designs without giving an extensive overview of all existing design concepts. In the past, an extensive overview of solid-rotor machines was presented in papers [3,4,15] and doctoral theses [9,16–18] in the past. Although these works provided valuable insights, they did not simultaneously and comprehensively address all the existing concepts or modeling approaches. Moreover, since their publication the stateof-the-art has further developed, driven by a growing interest in highspeed solid-rotor machines. This paper aims to fill in these gaps by presenting an extensive overview of the topic to date, covering all the major solid-rotor topologies, along with a detailed summary of their current applications, with a unique overview of the existing modeling techniques comparing the different correction factors that were mostly introduced separately in the past. The modeling of solid rotor electrical machines is different to the usual practice for conventional machines with a laminated rotor. The solid-rotor is not only part of the magnetic circuit through which magnetic flux flows, it also serves as a conductor of electrical current. Due to the non-linear rotor material properties, calculating the electromagnetic characteristics of an electrical machine with a solid-rotor is a complex task. The design of such a machine must take into account its multi-physical model, which has been the subject of a number of publications [19–21]. These models of high-speed machines typically include electromagnetic, thermal, and mechanical calculations. Electromagnetic calculation and design are among the first steps in the multi-physics design of high-speed electrical machines [22]. A list of requirements and a suitable machine topology are usually known or selected beforehand. For proper electromagnetic calculation, the precise temperature of the machine, especially of stator and windings, must be known. This calculation is therefore performed together with the thermal calculation in several iterative loops. Subsequently, the stress distribution in the rotor cross-section is usually verified and the rotor dynamic analyzed to determine how close to the critical speed the machine will be operating. The temperatures must also be known for mechanical analysis, since – depending on rotor topology – thermal expansions can influence the stress magnitude inside the rotor structure. If, the requirements for the machine cannot be met International Journal of Electrical Power and Energy Systems 166 (2025) 110520 2 V. Bilek et al. Fig. 1. Simplified multi-physics iterative procedure for designing a solid-rotor IM. in this process, then the machine must be redesigned or a different topology selected for the rotor. The iteration over these calculation steps is shown in Fig. 1. This review paper addresses the fact that solid-rotor IMs have advanced considerably, as evidenced by the significant number of scientific publications that have dealt with them in recent years. Based on these publications, a detailed summary of existing designs and associated modeling techniques is given. For each key modeling technique (electromagnetic, thermal and mechanical calculation), this article provides a comprehensive review of commonly used methods for effective calculation and design of solid-rotor IMs. This review is structured as follows: First, an overview of typical solid-rotor applications is given. This is followed by an overview of possible rotor topologies, including a discussion of their advantages and disadvantages from an interdisciplinary point of view. The subsequent sections describe electromagnetic, thermal, and mechanical modeling, with an emphasis on the characteristics of solid-rotors, which differ from those of laminated ones. The final section summarizes key messages. 2. Overview of solid-rotor IM applications As pointed out in [23], due to continuous developments in the field of power electronics, power inverters, manufacturing methods, and materials [24,25], use of high-speed solid-rotor electrical machines in high-speed applications has increased rapidly in the last decade. In general, the design of high-speed machines is very complex, and thus usually requires multidisciplinary optimization [23]. The high-speed machines are classified according to the circumferential speed of the rotor. In some industrial high-speed applications, electrical machines have directly replaced existing mechanical systems (composed of a gearbox and a low-speed electric machine) [3], while in others they complement existing mechanical systems. The general advantages of combining a high-speed machine with power inverters are the overall compactness, lower weight, and higher reliability of the systems, compared to the combination of a low-speed machine with a gearbox. Due to the simplicity of their manufacture, low cost, high accuracy, and rotor robustness, high-speed solid-rotor induction machines (HSSRIMs) are a very popular choice for high-speed applications [26]. Based on [23], Table 1lists the key applications for high-speed electrical machines. This paper covers mainly the most important application areas: oil, gas and air compressors and turbines, flywheel energy storage system, high-speed spindles, and turbomolecular pumps. Table 1 Power and speed requirements of selected applications, according to [23]. Application Power Speed Oil & gas 3–15MW 5–15 krpm Spindles 300W–60 kW 15–300 krpm Turbocharger 1–3 kW/10kW 150 krpm/80 krpm Air compressor 4kW–500 kW 15–80krpm Micro-turbines 30–400kW 15–120krpm Turbomolecular Pumps 50W–3 kW 70–100 krpm Fig. 2. Conventional compressor (a) and integrated compressor (direct drive) (b), according to [3]. Fig. 3. High-speed high-power IM for an integrated compressor. Cut view of a squirrelcage rotor (a) and detail of a high-speed laminated rotor [27]. 2.1. Gas compressor and turbine applications Gas compressors and turbines are generally needed in the chemical, oil, and gas industries, mainly for gathering, transmission, and processing the gas downstream [3]. Specifically for pressurized air or gas, compressors are used in many industrial applications including pneumatic actuators, sandblasting, machining, fermentation, instrumentation and, air polishing [19]. In recent years, due to advances in of power electronics [28], the combination of power inverters with highspeed electric machines has started to replace conventional machines with gearboxes. The clear advantages of this solution [29–31] are higher power density and efficiency, smaller size, and lower weight of the entire drive. In addition, the drive is oil-free, highly reliable, and low-maintenance [32]. Both drive types are illustrated in Fig. 2. To improve performance, lifespan, and efficiency of the compressor drive, use of active magnetic bearings is recommended [19]. IM are widespread in International Journal of Electrical Power and Energy Systems 166 (2025) 110520 3 V. Bilek et al. Table 2 Speed in typical milling applications, according to [41]. Applications Speed Metal 4500–12 000rpm Stones 8000–12 000rpm Glass/marble 8000–14 000rpm Wood 18 000–25 000rpm Aluminum 30 000–40 000rpm this industry, ranging from 3 MW to 15 MW with corresponding speeds of 20 krpm to 5krpm [23,33]. Fig. 3shows a high-speed high-power IM for an integrated compressor. 2.2. Flywheel energy storage systems Flywheel energy storages systems (FESSs) use a rotating flywheel to store mechanical energy which is then converted into electrical energy by an electrical machine. The electrical machine manages the transition from mechanical to electrical and back to mechanical energy [34]. A FESS stores energy as rotational energy in a rotating mass. The amount of energy stored depends on the inertia and speed of the rotating mass. To eliminate energy loss due to air friction, the flywheel is placed within a vacuum [35]. Ideally, it is suspended on active magnetic bearings [36] for stable operation and continuous rotation without added power and with minimal energy loss. FESSs can be classified, according to [37] into high-speed (10 k– 100 krpm) and low-speed (<6 krpm). Low-speed FESSs usually have large diameters and low power and energy densities [3]. High-speed FESSs have smaller diameters and higher power and energy densities, but, their power rating is typically constrained by cost considerations. High-speed FESSs can be up to five times more expensive than lowspeed FESSs [34]. For FESSs, either synchronous or IMs are used. IMs are the best choice for high-power applications, as they offer high torque, high reliability, and high robustness [38]. Since no electromagnetic losses caused by spinning are present, they are a great option for low-loss FESSs. FESSs are becoming a viable alternative to traditional chemical battery systems. According to [39], their advantages include higher energy storage density, lower risk of overcharge/over-discharge, easy measurement of the level of charge remaining, operation over a wide temperature range, longer lifespan, and environmental friendliness. They are suitable for switching between medium to high powers (kW to MW) for short periods of time with high energy efficiency (>85%) [40]. Generally, FESSs also allow a very high number of charge/discharge cycles (hundreds of thousands), independently of the temperature and the level of discharge. Their typical lifetime exceed 20 years, without negative environmental consequences. However, they can store energy for only a couple of hours at most [35]. Fig. 4shows schematically and photographically a typical FESS and its components. 2.3. High-speed spindle applications Conventional low-cost high-speed spindles use belt drives, which limits their maximum speed [3,23]. The ever-increasing demand for higher speed and speed control, low vibration levels, and higher power density and efficiency of this application has resulted in the adoption of high-speed machines. Speed limit and power range [44] of spindle applications vary considerably, ranging from 9krpm to 180k rpm, with corresponding approximate power from 24kW down to 1 kW. Effort is ongoing to increase the operating speed of high-speed spindle applications [41,45]. There are three main spindle applications [3]: milling, grinding, and drilling. The maximum spindle rotational speed for milling applications depends on the type of material processed, as shown in Table 2. The rotational speed is higher in grinding than in milling applications, with speeds up to hundreds of thousands of revolutions per minute [41]. Design examples of high-speed IMs with a squirrel cage were given in [46,47]. An example spindle is shown in Fig. 5. Fig. 4. Structure and components of a flywheel (a) (according to [42]) and flywheel systems kinetic energy recovery system for a Formula (1) car (b) [43]. Fig. 5. High-speed and high-precision motorized spindle for precision machinery (Jingdiao) [48]. 2.4. Turbomolecular pumps The high-speed turbomolecular pump [49] is – similar to the turbo pump – a type of vacuum pump, for creating and maintaining high vacuum. The working principle is based on momentum transfer [50]. It has multiple stages, each of which consists of a fast-spinning rotor and stator blades. The rotor blades hit the gas molecules, which have a high probability of being transferred preferentially to the next rotor stage after passing through the following stator stage. The whole system works like a pump, putting energy into the gas and pushing the gas from the inlet to the exhaust in order to maintain high vacuum. These pumps constitute the most reliable and cost-effective means of producing high vacuum, they are portable and fast-starting, and they require little support [51]. For better performance, low maintenance cost and friction-less operation creating ultra-clean oil-free vacuum environments, turbomolecular pumps with magnetic bearings have been used in recent years [52,53]. They are ideal for use with vacuum technology in high-tech fields such as film deposition, semiconductor manufacturing, high-energy physics, optical/glass industry, mass spectrometry, general ultra-high vacuum research, and fusion technology [54]. International Journal of Electrical Power and Energy Systems 166 (2025) 110520 4 V. Bilek et al. Fig. 6. Cross-section of a high-speed turbomolecular pump [56]. Turbomolecular pumps are used to achieve very high vacuum conditions pressures as low as 10−10 mbar, with rotational speeds up to 100krpm and only a few hundred watts up to a few kilowatts of output power [49]. The most commonly employed type of machine for this application is the three-phase IM. However, its design is more complex than that of conventional machines, especially in relation to extremely problematic heat exchange because the driving machine is within high vacuum. The stator can generally be cooled by water, but the rotor is cooled only by thermal radiation, the thermal dissipation efficiency of which is low [55]. The torque ripple must be very small to minimize the risk of mechanical resonance of the rotating system. An example of a turbomolecular pump is shown in Fig. 6. 3. Overview of solid-rotor IM topologies HSSRIMs are usually made of a single piece of ferromagnetic material or are composed of ferromagnetic and non-magnetic materials. The main advantage of this machine is its ability to reach very high speeds thanks to robust rotor construction. In high-speed applications, centrifugal forces and circumferential speed are key factors determining the choice construction type. This type of rotor is usually preferred over a laminated rotor for its megawatt range power and higher rotational or circumferential speeds. According to [16], the advantages of HSSRIM as: •high mechanical integrity, rigidity, robustness, and durability, •high thermal durability, •ease of protection against aggressive chemicals, •high reliability, •easy, straightforward, and cheap manufacture in most cases, •easy scaling for wide power and speed ranges, and •low noise and vibration levels (in the case of smooth solid-rotors). Consequently, HSSRIMs are often preferred in industry for high-speed and high-power-density applications. Below, the most important rotor topologies that are commonly used or widely discussed are presented. All rotor topologies listed could potentially be combined. 3.1. Smooth solid-rotor IM The simplest solid-rotor construction is a smooth solid-rotor (SSR), according to [16,58]. SSRs are the most mechanically robust, but lack in electromagnetic performance. Slip and losses of this type of machine tend to be large, and usually the power factor is low due to poor flux penetration [15] into the rotor caused by high rotor slip and low electrical conductivity of the rotor material. As a consequence, SSRs exhibit high apparent resistance, low magnetization inductance, and an over-saturated surface of the solid-rotor, which results in extremely poor electromagnetic torque density [17]. An SSR has the advantages of outstandingly simple manufacture, low cost, and the best mechanical and fluid-dynamical properties for low air friction. Its design is illustrated in Fig. 7(a). To improve the electromagnetic performance and increase the torque density of the machine, non-magnetic high-conductivity end rings can be added to the SSR, as stated in [16,18,59]. These help to improve current flow in the solid steel, which tends to be aligned more into the axial-parallel direction, thus increasing the Lorentz force. It has been reported, [18,58,60], that a two-pole SSR equipped with copper end rings produces twice as much torque at a particular slip than an ordinary SSR. The rotor configuration is illustrated in Fig. 7(b). Another possible SSR modification is to use a copper-coated smooth solid-rotor (CCSSR), as shown in Fig. 7(c). A conductive layer envelops the outer surface of the rotor, spanning from one end ring to another, as proposed in [15,61]. The thickness of the copper layer may be constant along the length of the rotor or may be increased in the end rings region. If the copper layer is thicker at the end rings, it enhances the conductivity of the rotor, thus improving the flow of electrical current within the rotor. Several authors have suggested, that either a lowor a high-conductivity non-magnetic material should be used for the coating [18,58,60]. A highly conductive layer (i.e., copper) acts like both an infinite number of rotor bars and end rings. Due to its high conductivity, it is the main circuit path for the rotor currents. It also acts as a high-frequency shielding for air-gap harmonics and stator slot harmonics, which do not penetrate through the coating layer [57,59]. This helps to reduce rotor eddy currents in the iron and stator winding losses. 3.2. Axially slitted solid-rotor IM A much better alternative to SSRs in terms of electromagnetic performance is an axially slitted solid-rotor (ASSR) design a simple rotor modification achieved by axially slitting the cross-section of the rotor. The axial slits contribute significantly to forcing the fundamental component of the flux deeper into the solid-rotor [12,58]. Further, axially slitting the rotor decreases the low-frequency impedance of the rotor, which results in higher torque and increased efficiency. Additionally, it increases the high-frequency surface impedance of the rotor, which results in lower rotor eddy-current losses of undesired harmonics [62]. According to [17], the main disadvantages of axial slits in a solid-rotor are the partially compromised robustness of the ASSR at higher circumferential speeds and that the air friction of the rotating rotor is significantly increased. However, the axial slitting improves rotor cooling because its surface is increased. This topology is shown in Fig. 8(a). Selecting the right number of slits is crucial. To achieve smooth torque characteristics, the combination of stator slots and rotor slits [60] should be chosen carefully to avoid synchronous torque dips. To select the number of rotor slits, an analogous method for the selection of rotor bars for squirrel cage IM can be used here. According to studies [16,63], the machine output torque increases with increasing number of rotor slits, as illustrated by Fig. 9. Only even numbers of rotor slits were considered in [16] for minimization of unbalanced magnetic pull (which may be greater than the rotor weight) and mechanical vibration of the rotor. Despite the risk of highly unbalanced magnetic pull, the electromagnetic torque ripple decreases when an odd number of rotor slits is used. It has been suggested [64] that between 5 and 15 slits per pole pair is ideal. International Journal of Electrical Power and Energy Systems 166 (2025) 110520 5 V. Bilek et al. Fig. 7. Construction of a smooth solid-rotor for an IM: (a) simple smooth solid-rotor, (b) smooth solid-rotor with copper end rings, and (c) copper-coated smooth solid-rotor, according to [16,17,57]. Fig. 8. Construction of an axially slitted solid-rotor for an IM: (a) axially slitted solid-rotor, (b) axially slitted solid-rotor with slit of various depth, (c) axially slitted solid-rotor with copper end rings, and (d) axially slitted rotor coated with conductive/resistive material, according to [17]. Fig. 9. Per unit output torque as a function of the number of rotor slits, according to [16,63]. Two essential aspects are slit depth and width. The slit depth has a significant effect on machine performance [60], and a depth of 60% of the rotor radius would allow the best machine electromagnetic parameters to be produced. However, such depth compromises the overall integrity and robustness of an ASSR, possibly resulting in rotor destruction or reduced life span. Thus, depths between 40%–50% of the rotor radius are usually chosen [65], depending on size, machine circumferential speed, and rotor mechanical strength. The slit width should be as thin as possible, but it is also limited by the mechanical stress that is generated at the bottom of the slit corner and that concentrates around the inner corners of the rotor slits. Significant reduction in this mechanical stress can be achieved by rounding the edges at the bottom of the slits, but in practice this solution cannot be implemented for all applications. The optimal slit width that achieves the best electromagnetic machine parameters, according to [13,66,67], is between 9 and 15% of the rotor tooth pitch. However, the slit width depends heavily on the technological capabilities of the manufacturer. The narrowest slits produced and reported to date [67] amount to 13.47% of the rotor tooth pitch. As a general rule, rotor slits should be designed as deep and thin as possible. The ASSR can be slightly modified by varying the slit depths, as shown in Fig. 8(b). The motivation for such a design is to reduce the high saturation of the material at the narrowest point between the slits and potentially increase machine torque. However, this rotor topology is also limited by the maximum slit depth (40%–50%). In [17] ASSR with slits of various depths restricted by the maximum allowable depth was considered, but the electromagnetic performance of the machine did not significantly improve. As with SSRs, copper rings can be added to ASSRs, as shown in Fig. 8(c). This design combines all the advantages described in Section 3.1, with the added benefit of axial slits that help to align the current flow in the axis-parallel direction. Torque and efficiency of the machine are thus increased [68]. As for the SSR, either a highly conductive or a resistive nonmagnetic material can be added to the surface of an ASSR, as shown in Fig. 8(d). If an ASSR is coated with a highly conductive material (e.g., copper) [17], the machine exhibits the best electromagnetic performance with the added benefit of axial slits. If coated with a resistive non-magnetic material [69], the machine behaves similarly. The resistive coat creates high surface impedance and acts as a high-frequency shielding. Regardless of the coating, rotor losses are significantly reduced. The coating helps to reduce the penetration depth of higher harmonic components from the air-gap into the rotor and can, for International Journal of Electrical Power and Energy Systems 166 (2025) 110520 6 V. Bilek et al. Fig. 10. Construction of an axially slitted solid-rotor for an IM, with axial slits skewedin tangential and axial directions: (a) identical slit depth and (b) slits of various depths, according to [11]. Fig. 11. Construction of a shielded axially slitted solid-rotor for an IM, according to [12,13]. designs with an odd number of rotor slits, reduce transient unbalanced magnetic forces. Additionally, if a suitable material is used, the coating can help to mechanically reinforce the rotor and considerably reduces frictional mechanical losses, which are significant for ASSRs at high speeds. The greatest disadvantages are high price and difficult manufacture. The coating should be welded into the rotor material, and the weld should have a higher tensile strength than the coating material. One special ASSR modification is skewing of the axial slits in the tangential direction, as illustrated in Fig. 10(a) and presented in [11], where the electromagnetic torque was thus increased by 10.7%. Another possible improvement is to vary the axial slit depth as illustrated in Fig. 10(b), which was shown in [11] to improve the electromagnetic torque by 37.7%. Slit cuts skewed in the tangential direction can be cut deeper because higher rotor integrity is retained. A comparison of the effects of various slit skew angles (15 ◦, 30 ◦, 45 ◦, and 60 ◦) found that electromagnetic torque increases slightly with increasing skew angle, but that the torque ripple increases by 40.3% in the case of 30 ◦skew angle. Skewing the axial cuts by 30 ◦in the tangential direction and by 26 ◦in the axial direction significantly reduced the torque ripple to 1% of the machine output torque [11]. The advantage of this kind of rotor modification is improved machine torque, and the disadvantage is substantially increased manufacturing complexity. A rotor topology was designed that is based on all the designs described and combines their advantages: the shielded axial slitted solid-rotor (SASSR), illustrated in Fig. 11. An SASSR consists of an ASSR, copper end rings, and a copper coating [12,13] and resembles a squirrel-cage solid-rotor. Axial slits lower the low-frequency impedance of the rotor and increase the penetration of flux into the rotor; the copper coating on the rotor teeth acts like rotor bars; and end rings collect the induced rotor current. The coating on the rotor teeth also minimizes the impact of higher-frequency air-gap harmonics on the flux inside the rotor and helps to reduce eddy current losses. This machine exhibits the highest electromagnetic torque and reduced torque ripple and eddy-current losses. A comparison [13] of ASSR, CCSSR and SASSR using numerical calculations showed that the SASSR had the highest torque, power output, and electromagnetic efficiency. The starting torque was remarkably higher than that of the CCSSR. However, to the best of our knowledge this is a theoretical concept that has never been produced, tested, and measured. Fig. 12. Construction of axially slitted solid-rotor with radial grooves for an IM, according to [70]. 3.3. Solid-rotor IM with radial rotor surface grooves This is a special case of solid-rotor modification with shallow radial grooves on the rotor surface acting as a coating. The radial grooves increase the surface impedance of the rotor and act as a highly resistive material layer (or high-frequency filter), providing the same properties as a highly resistive coating. To achieve the desired effect, the inter-groove distance and the groove width in the active rotor length must be designed correctly. Their depth should be similar to the penetration depth for the high-frequency harmonic components of the current induced in the rotor. Proper design should reduce the apparent conductivity of the outer rotor layer by 80%–90%. According to [18,70], eddy current losses at the rotor surface and total eddy current losses in the rotor can be reduced by up to 60% and 30%, respectively. This simple modification can be used with any type of solid-rotor that does not contain coating material. Its main advantage is simplicity of production leading to greatly increased efficiency. In addition, this modification does not negatively affect the mechanical and thermal properties of the machine. An ASSR with radial grooves is shown in Fig. 12. 3.4. Squirrel-cage solid-rotor IM Squirrel-cage solid-rotors (SCSRs) are almost identical to conventional induction machines with squirrel-age rotors. The main motivation for using an SCSR is that conventional laminated rotors are in most cases unsuitable for high-speed applications. Since a laminated rotor cannot mechanically withstand high centrifugal and high tensile forces [71], use of a solid-rotor technology with bored holes/cut slots and copper bars inserted and the bars connected by a short-circuit copper ring is preferable. Most of the current is induced in the rotor bars, where a small part of the current flows through the rotor steel [10, 72]. Compared to other solid-rotor designs, eddy current losses are significantly reduced because of much lower resistance of the rotor bars relative to the resistance of the rotor steel [73,74]. Such a design has a lower slip and a lower power factor, and it has the highest efficiency of all IM topologies. The simplest SCSR design has rectangular open slots with a brazed squirrel cage (Fig. 13(a)). A slightly better design from a mechanical point of view is a SCSR with a squirrel-cage embedded by isostatic pressure (Fig. 13(b)). International Journal of Electrical Power and Energy Systems 166 (2025) 110520 7 V. Bilek et al. Fig. 13. Construction of a solid-rotor with squirrel cage for an HSSRIM: (a) rectangular open slots with brazed squirrel cage, (b) squirrel-cage embedded by isostatic pressure, and (c) round embedded copper bars inserted into a solid-rotor with drilled holes according to [14]. Table 3 An overview of possible and suitable materials for solid rotors, along with their characteristic properties. This table was taken from [25] with the permission of the authors. Name Material Tensile strength [MPa] Resistivity [μΩ cm] at 20 ◦C (at 200 ◦C) Saturation flux density [T] Imacro M 4CrMn16-4 1100 25 (37.3) 1.63–1.95 AISI430 X6Crl7 600 25.7 (38) 1.66–1.74 Maraging Fe-Ni-Co-Mo-Ti 2200 49 1.9 C15 – 440 15.9 (30) 1.9 Supermendur Fe-V-Co 800 40 2.4 S355 – 520 25.7 (38) 1.9–2.1 Fe-Ni Fe-Ni 545 48.2 1.5 Fe-Cu Fe-Cu – 11 1.6 ARNOKRO ME III Fe-Cr-Co – 69 – Vicalloy Fe-Co-V 880 61.5 1.3 HyMu 80 Ni-Fe-Mo 930 58 – Consument – 345 13 2.15 Vim Var (AISI M50) C-Si-Mn-Cr-Mo-V 315 13 2.15 MuShield – 620 58 1.9 AISI H13 (Orvar) Cr-Mo-V 1820 52 (63) 1.9 X20Cr13 – 880 55 (66) 1.7–1.9 MoCN315M 34CrNiMo6 345 19 (33) 1.97–1.99 The problem with SCSRs is that inserting the rotor bars into the solid-rotor increases the leakage inductance of the rotor considerably, which reduces the performance of the machine. If the rotor slots are closed (Fig. 13(c)), the steel bridges between the rotor bars are saturated, as described in [60], thus reducing machine efficiency. This can be solved by cutting an opening above the rotor slots, thereby increasing the mechanical frictional losses. Despite all its challenges, if an SCSR IM is properly designed, it exhibits the best electromagnetic performance of all rotor topologies described. The main disadvantage is that manufacture is complex and challenging. Despite its numerous advantages, SCSRs are not the preferred choice for high-speed applications. In [22,75], it was stated that the most critical part of the design is the connection between the rotor bars and the short-circuit ring. Due to its high operating speed (and the associated high centrifugal forces) and high temperature, proper overall electromagnetic, thermal and mechanical design of the machine is crucial. This applies generally to all rotor topologies mentioned above, but, this topology is the most complex of all in terms of construction. 4. Suitable materials for solid rotors Each solid rotor topology requires the careful selection of the correct iron material. Pure solid rotors come with the disadvantage of high eddy-current losses and low power factor in comparison to laminated rotors. These disadvantages can be partially overcome with the addition of a copper layer, copper ends, or even, in some cases, a squirrel cage. With a purely solid rotor, it is advisable to select a material that has good electrical conductivity that allows the flow of rotor currents without high resistance and allows the production of high machine torque at low slip. This can be greatly enhanced by equipping the solid rotor with copper end rings. When additional copper elements, such as a squirrel cage or copper layer are added, it is advisable to choose iron with higher resistivity to limit rotor joule losses. The maximum saturation value of the material is also crucial and has a significant impact on the design of the machine and its overall performance. Besides electromagnetic parameters, the mechanical properties of the material must also be taken into account, it must also meet the thermal requirements of the machine. In high-speed applications, the material should have a sufficiently high tensile strength to allow the rotor to withstand high circumferential speeds and meet the rotor dynamic requirements. Table 3contains a summary of selected materials and their basic properties that are suitable for use in solid rotors. Some of the materials listed have been used in case studies to verify their suitability. In study [24], five materials were selected and compared when used in an IM with a solid rotor and squirrel cage with a rated power of 1 MW and a rated speed of 12 000 rpm. A similar study [25] was carried using seven different materials used in an IM with an ASSR with a rated power of 180 kW and a speed of 10 100 rpm. The last material group used for solid rotors are copper alloys and their use in copper-coated SSRs. The copper coating is applied to the solid rotor by explosive plating. Modeling a copper connection on a solid rotor using numerical methods is very difficult, as in practice the connection may not be uniform and air pockets may form. In order to successfully apply a copper alloy in a solid rotor, some important mechanical criteria needed to be met during the practical tests carried out by the authors. In practice, the ideal height of the copper alloy should be around 3–8 mm with a maximum of 10 mm. The final criterion is the elongation value of both the copper alloy and iron of the solid rotor. The elongation should be at least 10%for both materials. Table 4lists all the suitable copper alloys that can be used for explosive plating. No single alloy has the ideal properties, alloy selection is a trade-off between electrical conductivity, tensile strength, and material cost. International Journal of Electrical Power and Energy Systems 166 (2025) 110520 8 V. Bilek et al. Table 4 An overview of possible and suitable copper alloys for explosive coating of solid rotor surfaces and their characteristic properties. The percentage value of electrical conductivity is related to the maximum conductivity value of pure copper, respectively 58 MSm−1. Name EN code USN code Tensile strength [MPa] Electrical conductivity [%] Cu-OF CW008A – 240–360 100 CuNi2SiCr CW111C C18000 650–780 40 CuCr1Zr CW106C C18150 400–600 75 CuCr1 CW105C C18200 400–500 80 GlidCop AL-15 – C15715 500–700 55 5. Electromagnetic modeling of solid-rotor IMs 5.1. Analytical methods Analytical design usually considers significant simplifications of the machine. In the past, an unsaturated rotor with constant permeability was considered, which yielded very poor modeling accuracy when compared to real machine measurements [76]. Researchers then started to consider the 3D nature of solid-rotors, thus increasing the overall accuracy of analytical methods [76]. Modern analytical methods consider, solid-rotor saturation [77], and some researchers even split the rotor into several layers to obtain more accurate results: For instance, in [76] three-dimensional linear methods were combined with the transfer matrix method. Though its results are good, this approach is relatively complicated and limited to SSRs only. Newer analytical methods, some of which were presented in [78–86], aim for even better and more accurate results, but they are all limited to SSRs. Much simpler and more flexible analytical methods are equivalent electrical circuits (EECs) for steady-state machine performance calculation, whose greatest advantage is ease of use. More recent methods proposed in [49,87] use T-shaped EECs, with empirical formulas to ensure the best results. In [88], additional parameters were presented for calculation, such as resistance and end-region leakage reactance of the solid-rotor. A special case of single-phase CCSSR IM analysis using an EEC was presented in [89]: Here, the machine was analyzed by means of d-q axes modeling, where each axis has its own EEC. Overall, analytical methods are simple and fast, but they either require significant simplifications, which have a negative impact on computational accuracy, or are limited to specific solid-rotor topologies; they even may not work for the same type of machine with a different power range. The challenges of constructing accurate analytical methods can be summarized by the formula for penetration depth: 𝑑=√2 𝜔p𝜎 𝜇,(1) where 𝜔pis the angular frequency of the penetrating field, 𝜎is the electrical conductivity, and 𝜇is the magnetic permeability of the material. The penetration depth (1) is characterized by three variables that vary over time and space and are difficult to describe analytically. Further, all three variables interact at least partially. In addition, conductivity 𝜎changes with temperature, which is problematic, because the rotor is not heated uniformly throughout its volume. The angular frequency 𝜔psignificantly changes the behavior of the machine at high frequencies. It also greatly affects flux-lines penetration into the rotor. The magnetic permeability 𝜇varies considerably within the material used and due to the nonlinear B-H curve. In addition, the magnetic characteristics exhibit hysteresis effects. Since the rotor material is differently saturated throughout its volume, analytical methods must take into account the non-linear nature of the material. An example of the relative permeability distribution is shown in Fig. 14. Further, the distribution of rotor losses is shown in Fig. 15. Most of the rotor losses are concentrated in the outer layer of the rotor and are caused by the higher harmonic components that originate in the Fig. 14. Distribution of relative permeability in a smooth solid-rotor (a) and in an axially slitted solid-rotor (b). Fig. 15. Distribution of eddy current losses in a smooth solid-rotor (a) and in an axially slitted solid-rotor (b). air-gap and directly affect the penetration depth of rotor surface. In general, a solid-rotor IM is a complex multiphysics problem that is difficult to describe purely analytically. Modern numerical methods are more effective in solving such challenging problems and are thus used more frequently. 5.2. Numerical methods The currently most popular modern numerical methods for calculating solid-rotor IMs are based on the finite element method (FEM), the results of which are in very good agreement with measured data. Their main advantage is that the calculations are non-linear, while their main disadvantage is that they are computationally time-intensive. Solid-rotor IMs are complex electromagnetic-mechanical systems for which 3D calculations are recommended to capture all aspects of the machine. However, 3D models may require a very fine mesh (at least in some regions, such as the air-gap of the machine and the saturated end regions of the solid-rotor), and for IMs a time-stepping transient analysis is needed [90,91]. On the other hand, corresponding models are very time-consuming to evaluate and require considerable computer memory. On the other hand, they are able to calculate with high accuracy all machine properties, including the passage of magnetic flux from one pole to another in 3D directions [92], the complex distribution of eddy currents, and changes in magnetic flux density distribution [93], taking into account the leakage inductance of the stator end windings in 3D space [94] and considering the leakage inductance of the solid-rotor ends [95]. Although these 3D models yield accurate results [96], they are too time-consuming for most applications [96]. To minimize run time, time-harmonic analysis can be considered, but this approach is usually is only suitable for synchronous machines and cannot calculate all relevant aspects of IMs. Analytical methods in combination with the 3D meshed reluctant networks method (RNM) were presented as possible International Journal of Electrical Power and Energy Systems 166 (2025) 110520 9 V. Bilek et al. individual nodes are connected by thermal resistances. Individual heat losses of the analyzed machine are in the LPTN represented by current sources, which is based on an electro-thermal analogy [1]. In fact, the real temperature field in the machine is replaced by only a few nodal temperatures, which leads to a very small number of equations to be solved and thus to very fast computations [154,155]. Note that these models also have drawbacks, which are mainly related to the problematic modeling of some resistances of the LPTN. For instance, the thermal resistances of convective heat transfer are in principle speed-dependent. For some simple cases, these dependencies can be derived using the fundamentals of heat transfer theory [156, 157]. However, most geometries and cooling conditions in electrical machines, especially in high-speed machines (for instance, due to the rotor slitting), are too complex, and the convection thermal resistances must be modeled using empirical relationships or using CFD analysis. LPTN models also use a number of simplifications, e.g., real complex geometries must usually be replaced by simpler ones. Due to its advantages, this method is widely used for thermal modeling of electrical machines, including high-speed SRIMs [113,120, 121,158], and has a great potential for incorporation into optimization processes and multi-physics modeling. Therefore, more details on this method are given below. 6.2.3. Thermal models utilizing finite element method As previously mentioned in the paper, the finite element method is a powerful and popular tool for electromagnetic calculations. This method can also be used for a thermal analysis [159]. FEM-based modeling is thus the third option for implementing an electrical machine thermal model. In general, the FEM-based thermal models serve to calculate the temperature field in the machine under study. Compared to models based on the use of LPTNs, the FEM-based models have several advantages. For example, they are able to simulate arbitrarily complex geometries. Since they provide a complete temperature field of individual machine parts, they can be used to detect temperature hotspots. However, the FEM-based models require almost all the same input parameters as LPTN thermal models, including the heat transfer coefficients, which must be calculated by means of empirical equations or obtained by CFD analysis (see Section 6.2.2). In terms of computational power requirements, FEM-based models are less demanding than CFD-based models, but still significantly more demanding than LPTNs. Examples of FEM-based thermal analyses can be found in [160–164]. These models are also frequently used to verify the simpler LPTN models; see, for instance [122,165–167]. Fig. 25 shows an example of calculating the temperature field in an SSR and an ASSR using FEM. The results are normalized for a clear comparison. Similar air-gap cooling conditions were considered in the calculations. In addition, heat dissipation from the side walls of the teeth was taken into account in the ASSR, which, despite slightly higher losses, results in lower overall temperatures compared to the SSR, as can be seen in the figure. 6.3. Thermal modeling using a lumped parameter thermal network Up to this point, we have provided a thorough overview of electrical machine cooling and thermal modeling techniques that can be employed in SRIM applications. Further, we presented specific approaches to implementing a thermal model of an electrical machine. As previously mentioned, the LPTN thermal models are characterized by low computational power requirements, which makes them suitable for inclusion in multi-physics analyses. This section therefore concentrates on these models, particularly on their basic principles and fundamental aspects of their creation. This information may also be useful when using specialized commercial software that employs LPTNs. [168]. Fig. 25. Normalized temperature distribution in: (a) an SSR and (b) an ASSR. For a given LPTN, the nodal steady-state temperatures are calculated as a system of linear equations, where the number of nodes in the LPTN is equal to the number of equations to be solved. For a steadystate analysis, the unknown nodal temperatures are thus obtained by solving the matrix equation 𝐓=𝐆−1𝐏,(45) where 𝐓is the vector of unknown nodal temperatures, 𝐏is the power source vector, and 𝐆is the thermal conductance matrix. The individual parameters in Eq. (45) must be created for the specific LPTN as described, for example, in [1]. Examples of complete LPTN models of SRIMs can be found, for instance, in [113,120], and [121]. The accuracy of the temperatures calculated using an LPTN thermal model depends on several factors, including the complexity of the specific LPTN. Furthermore, accurate knowledge of heat losses also plays an important role. Primarily, it is essential to accurately calculate the individual thermal resistances of the LPTN. Computation of these resistances, which are modeled as thermal resistances by conduction, convection, or radiation, depending on the actual heat transfer mechanisms in individual machine parts, is described below. 6.3.1. Conductive heat transfer In solid parts of electrical machines, heat is transferred by conduction, which is a heat transfer mechanism that occurs in a substance due to the existence of a temperature gradient [156,157]. The basic equation associated with heat transfer by conduction is the heat conduction equation. Solving this differential equation with specific boundary conditions yields the temperature field in the geometry under analysis. For a general three-dimensional case within a medium with orthotropic properties, the heat conduction equation for a steady-state analysis can be written in the following form [169]: 𝜕 𝜕 𝑥(𝜆x𝜕 𝑇 𝜕 𝑥)+𝜕 𝜕 𝑦(𝜆y𝜕 𝑇 𝜕 𝑦)+𝜕 𝜕 𝑧(𝜆z𝜕 𝑇 𝜕 𝑧)+𝑝gen = 0,(46) where 𝑇is the temperature, 𝑝gen is the rate of internal energy generation per unit volume, and 𝜆𝑥,𝜆𝑦, and 𝜆𝑧are, respectively, the thermal conductivities in the individual directions of the Cartesian coordinate system. Note that for many parts of electrical machines, it is more convenient to work with the cylindrical coordinate system instead. Another important equation in the field of heat conduction modeling is Fourier’s law by which heat transfer rates can be calculated. For the general three-dimensional case, heat transfer rates 𝑃x,𝑃y, and 𝑃z in individual Cartesian coordinates are calculated by [156]: 𝑃x= −𝜆x𝐴x𝜕 𝑇 𝜕 𝑥,(47) 𝑃y= −𝜆y𝐴y𝜕 𝑇 𝜕 𝑦,(48) 𝑃z= −𝜆z𝐴z𝜕 𝑇 𝜕 𝑧,(49) International Journal of Electrical Power and Energy Systems 166 (2025) 110520 16 V. Bilek et al. where 𝐴x,𝐴y, and 𝐴zare heat conduction areas normal to the 𝑥-, 𝑦-, and 𝑧-directions of the Cartesian coordinate system, respectively. By using a known temperature field and heat transfer rates obtained by means of Fourier’s law, thermal resistances by conduction can be derived and then used in an LPTN. The procedure for such a derivation can, for instance, be found in [1,170]. Note that the use of the heat conduction Eq. (46) leads to a very complex mathematical solution, and thus several simplifying assumptions are usually taken into account in LPTN models. Typically, it is desirable to simplify the multi-dimensional temperature field of some parts of the machine analyzed. This can, for instance, be done in the cases where the heat transfer in one or more directions is negligible compared to the heat transfer in the other directions. Such a simplification can usually be assumed, for example, for the stator iron of electrical machines due to lamination effects or, eventually, also due to poor cooling conditions on its end faces. Thus, a one-dimensional temperature field with heat transfer only in the radial direction can be considered in this part of the machine in many practical cases [171, 172]. However, there are cases in which the multi-dimensional temperature field cannot be simplified, and heat transfer in multiple dimensions must be considered. This also applies to the modeling of heat transfer by conduction in solid rotors where absence of lamination makes multi-dimensional temperature fields more likely. For these multi-dimensional cases, some simplifications are introduced in the calculations with LPTNs: The multi-dimensional conduction is solved as a combination of one-dimensional solutions in the corresponding directions. Approaches of this kind were presented for modeling the three-dimensional heat transfer in a general cuboidal element in [170] and for modeling the three-dimensional heat transfer in a general arcsegment element in [173]. In practice, these general elements, referred to as model bodies, are used for modeling the conductive heat transfer in individual machine parts due to the geometric similarity between these model bodies and real machine parts. Fig. 26 shows an example of a body that can be used for twodimensional heat transfer modeling in cylindrical machine parts, for instance, in the solid-rotor [121]. In particular, using this model body allows the heat transfers in the radial and axial directions to be solved separately. Fig. 26(a) illustrates all important parameters, such as dimensions, where 𝑅in is the inner radius, 𝑅out is the outer radius, and 𝐿is the length of the cylinder. The boundary conditions assumed are also indicated in the figure, where 𝑇r,in is the average temperature of the inner cylindrical surface, 𝑇r,out is the average temperature of the outer cylindrical surface, 𝑇a,left is the average temperature of the left end surface, 𝑇a,right is the average temperature of the right end surface, and 𝑃loss are the heat losses generated in the body. All parameters associated with the radial and axial directions have indices ‘r’ and ‘a’, respectively. The corresponding thermal network that enables calculation of multi-dimensional heat transfer separately in the radial and axial directions can be seen in Fig. 26(b). The individual resistances are calculated according to the following equations [121,153]: 𝑅1,r =1 4𝜋 𝜆r𝐿⎡⎢⎢⎢⎣ 2𝑅2 out ln (𝑅out 𝑅in ) (𝑅2 out −𝑅2 in)− 1⎤⎥⎥⎥⎦ ,(50) 𝑅2,r =1 4𝜋 𝜆r𝐿⎡⎢⎢⎢⎣ 1 − 2𝑅2 in ln (𝑅out 𝑅in ) (𝑅2 out −𝑅2 in)⎤⎥⎥⎥⎦ ,(51) 𝑅3,r = −[𝑅2 in +𝑅2 out − 4𝑅2 in𝑅2 out ln(𝑅out 𝑅in ) (𝑅2 out−𝑅2 in)] 8𝜋 𝜆r𝐿(𝑅2 out −𝑅2 in),(52) 𝑅0,a =𝐿 2𝜋 𝜆a(𝑅2 out −𝑅2 in),(53) Fig. 26. Model body for two-dimensional heat transfer modeling in cylindrical machine parts: (a) geometry with important parameters, (b) corresponding thermal network. Source: Adapted from [121,153]. 𝑅1,a = −𝐿 6𝜋 𝜆a(𝑅2 out −𝑅2 in),(54) where 𝜆rand 𝜆aare the thermal conductivities in the radial and axial directions, respectively. In general, these parameters may have different values: 𝜆r≠𝜆a. In the case of a solid-rotor, however, these thermal conductivities are typically equal: 𝜆r=𝜆a. 6.3.2. Convective heat transfer In a quiescent fluid, heat is transferred by conduction. However, in the presence of bulk fluid motion, heat transfer through the fluid is enhanced and heat is transferred by convection in this case [156]. Proper modeling of convection is essential in SRIMs because the rotor heat losses, which actually achieve significant values in these machines, are mostly dissipated by this mechanism. The rate of heat transfer by convection between a solid body and its surrounding fluid, for instance, between the rotor surface and the fluid in the air gap, is calculated by 𝑃=𝛼conv𝐴conv(𝑇surf −𝑇∞),(55) where 𝛼conv is the convection heat transfer coefficient, 𝐴conv is the surface area, 𝑇surf is the surface temperature of the solid body, and 𝑇∞is the free stream temperature of the surrounding fluid. Thermal resistance by convection is calculated by 𝑅=1 𝛼conv𝐴conv .(56) The surface area 𝐴conv can usually be calculated very easily and precisely from a particular geometry, which is also relatively simple in the case of SRIMs. However, determining the accurate value of the heat transfer coefficient 𝛼conv is challenging. This coefficient depends on several factors, such as the physical properties of the cooling fluid. The proper choice of cooling fluid can therefore significantly affect the effectiveness of convection cooling [126]. Further, the heat transfer International Journal of Electrical Power and Energy Systems 166 (2025) 110520 17 V. Bilek et al. coefficient heavily depends on the shape and location of the solid body relative to its surrounding fluid, and thus different equations need to be used to calculate heat transfer coefficients for different surfaces of the machine analyzed. For these purposes, empirical correlations based on dimensionless analysis are typically used [145]. Specific relations for calculating the heat transfer coefficient in the air gap can be found, for instance, in [121]. The literature gives formulas for both typical cases of air gap cooling, that is, with and without axial flow. An overview of other relations for calculating the heat transfer coefficient in the air gap was given in [174,175]. Some relations for heat transfer coefficients of other important machine parts can be found in [143,156,157,176], and [141]. 6.3.3. Radiation heat transfer Unlike heat transfer by conduction and convection, radiation heat transfer does not require a medium and can take place in a vacuum, since thermal energy is transferred via electromagnetic waves here. In most practical cases, radiation heat transfer is ignored, as it is negligible compared to conduction and convection [171,172]. However, there are some specific situations where radiation can play a significant role, for instance, when the machine is employed in aircraft and aerospace applications, where very low air density or even vacuum conditions may occur [177]. The amount of heat transferred by radiation depends on several factors, including temperatures, emissivities, shapes, and also the relative orientation of the surfaces between which heat is exchanged. Values of the emissivities for various materials and their surface finishings, as well as particular equations for calculating the amount of heat transfer depending on the various mutual configurations of the individual radiation surfaces, can be found in [156,157]. For example, the heat flow rate by radiation between the stator and the rotor can be modeled as a case of two long concentric cylinders, calculated as [119,157] 𝑃= 𝜎SB𝐴in(𝑇4 in −𝑇4 out) 1 𝜖in +𝑅in 𝑅out (1 𝜖out − 1),(57) where 𝑃is the heat flow rate between the inner and outer cylinders, 𝜎SB = 5.67 × 10−8 W m−2 K−4is the Stefan–Boltzmann constant, 𝑇in is the surface temperature of the inner cylinder, 𝑇out is the surface temperature of the outer cylinder, 𝜖in is the emissivity of the inner cylinder, 𝜖out is the emissivity of the outer cylinder, 𝑅in is the radius of the inner cylinder, 𝑅out is the radius of the outer cylinder, and 𝐴in = 2𝜋 𝑅in𝐿is the surface area of the inner cylinder, where 𝐿is the length of the cylinders. Including radiation heat transfer in an LPTN thermal model requires knowledge of the thermal resistance by radiation. To obtain this resistance, the equation for calculating the radiation heat flow rate must first be rewritten as 𝑃=𝛼rad𝐴in(𝑇in −𝑇out),(58) where 𝛼rad is the radiation heat transfer coefficient. For the case of two long concentric cylinders, comparing Eqs. (57) and (58) yields for the radiation heat transfer coefficient: 𝛼rad =𝜎SB 1 𝜖in +𝑅in 𝑅out (1 𝜖out − 1)(𝑇4 in −𝑇4 out) (𝑇in −𝑇out).(59) Finally, the thermal resistance by radiation can be calculated by 𝑅=1 𝛼rad𝐴in .(60) Note that the temperatures in the equations above are, in fact, unknown temperatures calculated by the thermal model. Since this requires iterative computation, the inclusion of radiation in the thermal model should always be considered carefully. Fig. 27. Example of a machine with through-ventilated cooling: (a) coolant flow path, (b) temperature profile. Source: Adapted from [121]. 6.3.4. Modeling of coolant flow As previously mentioned, solid-rotor induction machines require effective dissipation of heat losses generated in the rotor. In these machines, maintaining rotor temperatures at acceptable levels is often achievable only by using cooling strategies that include direct heat dissipation from the rotor to a cooling fluid flowing in its proximity, for example, through the machine’s air gap (Figs. 24 and 27). A comprehensive overview of coolant flow modeling can be found, for example, in [1,178]. The most important aspects are summarized below and may apply not only to the specific air-gap cooling configuration considered here but also generally to any cooling topology with circulating coolant, for example, to water jacket cooling and shaft cooling. An example of a machine with through-ventilated cooling is shown in Fig. 27(a). Due to its simplicity, this setup is convenient for explaining the fundamental concepts of coolant flow modeling, and was also employed for this purpose, for instance, in [121]. Fig. 27(b) shows a typical temperature profile of the coolant as a function of the position in the coolant flow path. As the coolant passes through the machine, it absorbs heat losses that cause the coolant to heat up from its input temperature 𝑇in to the output temperature 𝑇out. The total temperature rise of the coolant depends on the volume flow rate, which can be calculated by 𝑞=𝑃tot 𝜌𝑐p𝛥𝑇tot =𝑃tot 𝜌𝑐p(𝑇out −𝑇in),(61) where 𝑞is the volume flow rate of the coolant, 𝑃tot are the total heat losses absorbed by the coolant, 𝛥𝑇tot is the total temperature rise of the coolant, and 𝜌and 𝑐p, which describe the coolant’s physical properties, are the density and the specific heat capacity at constant pressure, respectively. The coolant temperature is usually analyzed in several important machine parts, and therefore the temperature profile (Fig. 27(b)) is divided into several corresponding sections. In each of these sections, a linear temperature rise is assumed. The temperatures 𝑇1,𝑇2, and 𝑇3 International Journal of Electrical Power and Energy Systems 166 (2025) 110520 18 V. Bilek et al. correspond to the average temperatures of the coolant in the left endwinding region, in the air gap, and in the right end-winding region of the machine, respectively, while the temperatures 𝑇in,1,𝑇end,1,𝑇in,2, 𝑇end,2,𝑇in,3, and 𝑇end,3 are input and output coolant temperatures of the corresponding sections. The linear temperature rise in the 𝑖th section of the coolant flow path is described by 𝛥𝑇𝑖=𝑇end,𝑖 −𝑇in,𝑖 =𝑃loss,𝑖 𝜌𝑞 𝑐p ,(62) where 𝛥𝑇𝑖,𝑇in,𝑖,𝑇end,𝑖, and 𝑃loss,𝑖 are the coolant total temperature rise, the coolant input temperature, the coolant output temperature, and the heat losses absorbed by the coolant, respectively, corresponding to the 𝑖th section of the coolant flow path. The average temperature in any specific section is equal to half of the total temperature rise in that section plus the input coolant temperature in that section, as can be seen in Fig. 27(b). This statement can be expressed as 𝑇𝑖=𝑇in,𝑖 +𝛥𝑇𝑖 2=𝑇in,𝑖 +𝑇end,𝑖 −𝑇in,𝑖 2,(63) which, like Eq. (62), is generally applicable. Using Eqs. (62) and (63) yields the average coolant temperatures in the individual sections (Fig. 27): 𝑇1=𝑇in +𝑅q𝑃loss,1,(64) 𝑇2=𝑇in +𝑅q(2𝑃loss1 +𝑃loss2),(65) 𝑇3=𝑇in +𝑅q(2𝑃loss1 + 2𝑃loss2 +𝑃loss3),(66) where the resistance 𝑅qis calculated as 𝑅q=1 2𝜌𝑞 𝑐p .(67) Note that the temperatures 𝑇1,𝑇2, and 𝑇3are nodal temperatures of the corresponding LPTN of the machine analyzed and are calculated concurrently with all other nodal temperatures. Further, when an LPTN contains nodes that represent a coolant, the matrix Eq. (45) must be extended by matrices corresponding to the coolant nodes, as described in [1]. Finally, note that the required volume flow rate of the coolant must be ensured by adequate fans, pumps, etc. For this purpose, the hydraulic circuit of the machine must be analyzed [141,145,179–182]. 7. Mechanical modeling of solid-rotor induction machines In general, mechanical calculations are used to estimate rotor safety, determine critical speeds and calculate bearing forces. While there are usually no major complications with low-speed machine design, for the HSSRIM the mechanical design can be a challenging. Furthermore, the requirements for an electromagnetically and mechanically optimal design are often conflicting. For example, from a mechanical point of view, it is preferable that the slits of an ASSR design (Fig. 8) be as shallow and wide as possible, with a large fillet at the bottom, and for SCSR (Fig. 7) the bars should be as deep beneath the surface as possible. This generally conflicts with electromagnetic requirements, so some compromise solution must be found. The mechanical stress in the rotor must be below the yield strength, and usually some safety factor is considered. From a rotordynamic point of view, there are requirements concerning the level of vibrations, because they generate additional forces on the bearings, produce excessive noise, and can cause the entire machine to resonate. Vibrations are caused either by insufficient rotor balancing or by the operating speed being close to the critical speed. The latter can be determined based on the eigenfrequency of the first bending mode shape. It is evident that short rotors therefore have a higher critical speed. The operating speed of an HSSRIM is often near the critical speed. Hence, the critical speed must be determined with sufficient accuracy during the design phase. There are several types of loads that are considered in mechanical calculations, including those resulting from rotation, temperature, rotor unbalance, unbalanced magnetic pull, and external forces. Depending on the circumstances, not all types of loads contribute significantly to rotor stress. The importance of a particular load must be determined in order to decide whether it is to be included in the analysis. 7.1. Stress calculation Well-known stress formulas for a rotating cylinder and for bending or torsion loading can be used however, these equations are derived for a smooth rotating cylinder, and incorporating deviations from this shape can be very challenging. The main advantage of the analytical solution is its short evaluation time, which is advantageous when iterating the stress calculation in the optimization loop. For more complex designs, such as ASSR and SCSR, it is often more effective to use FEA. In this case, again, a 2D model is frequently used (e.g. [183], and [184]), which has the advantage of shorter computation time and the disadvantage that only the stress in a cross section sufficiently far from the ends of the rotor can be determined. Calculating the stress in the rotor ends requires a 3D model. It is efficient to apply symmetry conditions and perform the calculation on a cyclic section only, as done, for example, in [75]. This significantly reduces the computational complexity of the model, which is especially desirable when a nonlinear problem must be solved, for instance, for press-fitted rotor parts or a fit with clearance (e.g., SCSR with the bars inserted into holes). 7.1.1. Load from rotation The rotational load is due to centrifugal forces and generates stress that increases quadratically with increasing machine speed. The analytical approach is sufficiently accurate for SSRs (Fig. 7), where the stress in the middle section can be calculated using differential equations obtained, for instance, in [185]. The equations are derived for the plane strain condition, that is, for a long rotating cylinder (Fig. 28). They describe the stress components 𝜎r,𝜎t, and 𝜎zas functions of 𝑟. Based on [185], these stress components can be calculated respectively by 𝜎r=3 +𝜇 8𝜌𝜔2(𝑅2 out −𝑟2),(68) 𝜎t=3 +𝜇 8𝜌𝜔2𝑅2 out −1 + 3𝜇 8𝜌𝜔2𝑟2, and (69) 𝜎z=𝜇(𝜎r+𝜎t).(70) In these equations, 𝜇is Poisson’s ratio, 𝜌is the rotor density, 𝜔is the rotor angular velocity, 𝑅out is the rotor outer radius and 𝑟is the variable distance in the radial direction. The stress value that is subsequently compared with the yield strength is called an equivalent stress. The most commonly used type of equivalent stress, von Mises stress (𝜎vM), can be calculated by Eq. (71) according [186]. To guarantee safe operation, the maximum value of the equivalent stress must be below the yield strength, which is a material mechanical property. 𝜎vM =√(𝜎r−𝜎t)2+ (𝜎t−𝜎z)2+ (𝜎z−𝜎r)2 2. (71) A typical plot of the stress component in the rotating solid cylinder is shown in Fig. 29. Note that the stress is normalized, which means that all values are relative to the maximum value of the equivalent stress 𝜎vM. From Eqs. (68) through (71) it is clear that the maximum value of the equivalent stress is at the point where 𝑟is infinitely close to zero, which is at the axis of rotation. Each geometric transition acts as a stress concentrator, including the transition at the connection of shaft and rotor. To obtain the stress in this region, the corresponding stress components must be multiplied International Journal of Electrical Power and Energy Systems 166 (2025) 110520 19 V. Bilek et al. Fig. 28. Cross section of the long rotating cylinder — main dimensions. Fig. 29. Normalized stress components in a rotating solid cylinder (based on Eqs. (68) through (71)). by the stress concentration factor. Stress increase due to geometric transitions has been investigated for a very long time, and the values of the stress concentration factors were summarized according to form and shape of the stress raiser, load case and structural member type, for example, in [187]. Similarly, but with some uncertainty, the maximum stress value at the bottom of the notches can be determined, for ASSRs. The effect of rotor slit dimensions on the stress was investigated, for instance, in [183]. The slots in SCSRs are also stress concentrators. In this case, additional forces from the inserted bars act inside the holes. As mentioned, for more complex designs, such as ASSR and SCSR, it is often more convenient to use FEA. A comparison of stress in a SSR, copper-coated SSR, SCSR with copper bars inserted in drilled slots, and ASSR is shown in Fig. 30. The stress level is normalized, which means that the contours in the figure represent multipliers of the reference value, which is defined as the maximum equivalent stress (𝜎vM) in a smooth rotating cylinder. There is a significant increase in stress at the base of the slits in the ASSR design and around the holes in the SCSR design. Further examples of equivalent stress calculations can be found in [184], where in addition to stress from rotation the thermal stress due to the different thermal expansions of materials in the contact was included and the stresses induced by any clearance or interference fits were considered. Similar analyses were presented in [75,188]. Torsional stress is usually not calculated, since the most analyses assume the constant machine speed and the torque, especially in HSSRIMs, does not provide a significant load. However, in cases where a high moment of inertia body (e.g. a flywheel) is attached to the rotor, significant torsional stress may arise during acceleration and deceleration of the machine. This stress must be included in the calculation of the equivalent stress (𝜎vM). A simple analysis of the torsional stress Fig. 30. Normalized stress from rotation. Contours represent multipliers of the maximum equivalent stress (𝜎vM) in the smooth cylinder: (a) SSR, (b) CCSSR, (c) SCSR, (d) ASSR. (The designs are described in Section 3). for squirrel-cage rotors was performed in [189]; the model analyzed consisted of a motor, a shaft and a disk-shaped flywheel. 7.1.2. Temperature load Temperature stress is caused by thermal expansion of the material. In high-speed machines that operate at high temperatures, this stress can easily exceed the yield strength. The basic relationship of the difference in length with temperature change is given by: 𝛥𝐿 =𝐿0𝛼 𝛥𝑇 ,(72) where 𝛥𝐿 is length increment, 𝐿0the initial length, 𝛼the coefficient of thermal expansion, which is a material property, and 𝛥𝑇 the temperature increment. If the rotor is assembled from materials that differ in thermal expansion (SCSR with the bars inserted into holes), the tolerances between the individual parts should also be taken into account. Heating and cooling may reduce the clearances or cause contact loss, which may cause problems during operation. The interfaces between different materials (SSR with copper end rings, copper coated SSR, SCSR) should also be analyzed carefully. This problem is not limited to rotors that contain multiple materials with different temperature expansions. Thermal expansion can also create significant stresses at geometric transitions (e.g., at the bottom of the slits in an ASSR). An example of the stress caused by the combination of temperature load and rotation is shown in Fig. 31. The stresses in the figure have been normalized in the same manner as in Fig. 30. It can be seen that the stress generated by thermal loading can be very high, especially in the interfaces between two materials and at locations with non-smooth geometry. It can easily exceed the yield strength. Thermal stress was analyzed in [75], where the clearance was optimized to minimize the thermal stress, and in [184], where a 2D model of the rotor was loaded by rotation and by temperature at the lowest and highest levels sequentially. 7.1.3. Unbalance Unbalance is an inherent property of any rotor. All rotors should therefore be balanced before the machine is assembled. During the International Journal of Electrical Power and Energy Systems 166 (2025) 110520 20 V. Bilek et al. Fig. 31. Normalized stress from rotation and temperature loads. Contours represent multipliers of the maximum equivalent stress (𝜎vM) in the smooth cylinder: (a) SSR, (b) CCSSR, (c) SCSR, (d) ASSR. (The designs are described in Section 3). Fig. 32. Balancing an unbalanced rotor. balancing process on balancing machines, masses are added or removed in the balancing planes until the rotor inertia forces are eliminated (Fig. 32). The rotor is considered balanced when the residual unbalance is below the limit given by the relevant standard. A description of the balancing process can be found, for example, in [190,191]. Unbalance may also occur during operation, for example, due to sliding of a part of the compound rotor (SCSR with the bars inserted into holes), corrosion, shaft bend, or mechanical wear of the rotating parts. In the case of excessive unbalance, the forces acting on the bearings are increased, resulting in excessive vibration and noise. Unbalanced rotor vibration was discussed in [192], where the authors measured both the vibration during rotor start-up (transient response) and the steady vibration at a constant speed. The simulation of the start-up of an unbalanced rotor mounted on flexible supports described by stiffness and viscous damping was discussed in [193], where the authors used a simple model with three degrees of freedom to investigate the planar oscillations of the rotor. They showed that the inertial forces acting on an unbalanced rotor produce a torque opposing the driving torque. 7.1.4. Unbalanced magnetic pull (UMP) If rotor and stator are concentric, the radial magnetic forces are in equilibrium, but a problem arises if the air gap is not homogeneous. Such an inhomogeneity can not only be caused by rotor misalignment, but also by rotation of an unbalanced rotor, especially near the critical speed. In this case, the rotor structural displacement increases, and thus the air gap becomes non-uniform, which results in increased force in the direction of the deformation on the side where the air gap is smaller. 7.1.5. External loads External forces, such as those due to the gear or belt transmission and gravitational forces, can act on the rotor to increase the load on the bearings. Considering these forces when selecting bearings is crucial. The direction of these forces is stable in a non-rotating coordinate system, but on a rotating shaft they appear as periodic loads that cause cyclic stress. A large amplitude of cyclic stress can cause material fatigue. The external load acting on the rotor may also include forces that are due to misalignment of coupled machines (as shown, for example, in Fig. 2). There are two basic types of misalignment: angular misalignment, where the shaft axes of the two machines meet at an angle relative to each other, and parallel misalignment, where the shaft axes of the two machines are parallel, but have an offset. These can be sources of vibration and appear in the frequency spectrum at typical multiples of the rotational frequency. More detailed information can be found, for example, in [194]. 7.2. Critical speed calculation The critical speed is the machine rotational speed at which the rotor vibrations significantly increase. It is given by the natural frequency (also known as eigenfrequency) of the first bending mode shape, where the frequency units (e.g., Hz) can simply be converted to units of speed (typically, rpm). The natural frequencies and mode shapes can be determined by solving the equation 𝐌 𝐱+𝐂 𝐱+𝐊𝐱 =𝟎, (73) where 𝐌,𝐂and 𝐊are the mass matrix, damping matrix and stiffness matrix, respectively. It can be seen that the results depend only on rotor geometry, stiffness, and damping characteristics. Any phenomena that affect these matrices should be taken into account in the calculation of critical speeds. Once the matrices have been specified, the mode shapes and eigenfrequencies can be determined as described, for instance, in [195]. Figs. 33(a) and 33(b) show an examples of the first bending mode shapes of an SSR, and an ASSR, respectively. Although the shapes look similar, the natural frequency (and therefore the critical speed) of the slitted rotor is higher (in this case by about 5%) because the axial slits do not significantly modify the bending stiffness, but decrease the rotor mass. In general, reducing mass increases the natural frequency, and reducing stiffness reduces the natural frequency. Analytical methods are adequate for calculating the critical speed for simple arrangements (SSR or ASSR). For more complex designs, it is more convenient to use FEA. Determining the critical speed for SCSR designs, where the bars are inserted into slots is somewhat problematic because, due to manufacturing tolerances, the extent of clearance or overlap between bar and hole are not accurately known, and additionally they are affected by temperature. It is therefore unclear how much the bars contribute to the final bending stiffness of the rotor. The bars are often conservatively assumed to add only mass and not bending stiffness. Since the frequency of the first bending mode shape must be calculated, a model of the full rotor must be used. The stiffness and damping matrices 𝐊and 𝐂usually include the properties of the coupling elements (e.g., bearings and seals), which are often speed-dependent. The terms of the matrices can therefore be affected by the rotational speed. The rotor-bearing system behaves like a series of springs. It is known that the resulting stiffness of a series of springs is lower than that of individual springs. The critical speed of such a system is thus lower than in the case of rotors in rigid International Journal of Electrical Power and Energy Systems 166 (2025) 110520 21 V. Bilek et al. Fig. 33. Example of the first bending mode shape for SSR design (a) and ASSR design (b). Fig. 34. Campbell diagram showing the procedure for determining the critical speed. supports. In [196] the bearing stiffness in relation to critical speed was investigated. In cases where the natural frequency depends on rotor speed, the critical speed is usually determined from a Campbell diagram, sometimes (e.g. in [197]) called a natural frequency diagram. An example of a Campbell diagram is shown in Fig. 34. The critical speed corresponds to the intersection of the natural frequency curve with a line that describes the rotational frequency. Vibrations caused by the inherent unbalance of the rotor during start-up at supercritical speeds were calculated in [198], where the vibration amplitude at various start-up speed profiles was determined. Many aspects that influence the critical speed are often not incorporated into the computational model; these include the gyroscopic effect, investigated in [199], asymmetric stiffness properties of the bearing or bearing supports, investigated in [200], or speed-dependent bearing properties, studied in [196], and a shaft with dissimilar moments of area (asymmetrical shafts), evaluated in [197,201]. In the last of these, the matrices 𝐌,𝐂and 𝐊in Eq. (73) depend on the angle of rotation. This was evaluated, for instance, in [202]. Another phenomenon that affects the critical speed is the UMP, described in Section 7.1.4. Since its inclusion in the calculation decreases the critical speed obtained, the UMP should be taken into account especially if the operating speed is close to the critical speed. A corresponding analysis was provided, for example, in [203], where the impact of UMP on the critical speed was evaluated by means of a transfer matrix method and FEA. A UMP analysis was also presented in [196], where the effect of rotor eccentricity on UMP was studied. Simulation and experimental verification of the UMP were reported in [204]; In the experiment, the rotor static eccentricity was adjusted by adding a shimming plate between housing and motor carrier. It was shown in [205] that it is difficult to gain insights into individual effects based on a very complex model. It is more informative to analyze a simple model that includes all factors of interest. 8. General overview of rotor topology characteristics The rotor topologies described in this paper are summarized in Table 7, accompanied by a comparison of their key characteristics. Rotor topologies with a squirrel cage or copper coating provide the best electromagnetic performance but are challenging to manufacture. Moreover, solid rotors with squirrel cages demonstrate some of the worst mechanical properties and poor rotor robustness. These shortcomings are not present in SSRs and ASSRs. However, machines with these rotors exhibit significantly poorer electromagnetic performance. The performance of the machine can be improved by equipping the solid rotor with a copper end-ring or, in the case of an ASSR, by skewing the slits. Despite these modifications, it is still not possible to achieve the same performance as for machines with a squirrel cage or a copper coating. It cannot be clearly determined which rotor topology is the best. The manufacturing of specific rotor topologies can pose technological and financial challenges, which arise from the desirability of the excellent electromagnetic parameters and favorable mechanical and thermal properties inherent in these designs. The normalized electromagnetic characteristics of the four base topologies for solid rotor induction machine are compared to those of the laminated cage squirrel cage induction machine, and shown in Fig. 35. In terms of torque vs. slip (Fig. 35(a)), the simple SSR machine has, as expected, the worst torque vs. slip characteristic. In contrast, the slitted ASSR machine has approximately three times the torque value when compared to the simple SSR. A laminated and solid rotor machine with a squirrel cage has almost the same nominal and starting torque. The greatest deviation in the torque can be seen from breakdown to the starting torque. The copper-coated SSR machine has much higher nominal torque than the other SSR, and the highest starting torque of all the machines. Due to the higher leakage reactance in this machine the breakdown torque is lower than squirrel cage rotors. This may change with different a design or power rating of the machine. Regarding current vs. slip (Fig. 35(b)), the situation is very similar. The simple SSR and slitted ASSR have the lowest overall values of phase current, followed by the copper-coated SSR machine which has the highest nominal phase current value of all machines, both in the area of low slip (highest no-load current) and machine start-up. This is mainly due to the height of the copper layer. Here, the copper layer is relatively high which requires a high magnetizing current to compensate for the air-gap length and copper layer. The laminated and solid rotor within squirrel cage machines, exhibits almost identical phase current values and the laminated rotor machine shows the highest values of all machines. In conclusion, these electromagnetic characteristics are only informative and do not represent the absolute character of the studied machines. In specific cases and with machine designs utilizing specific rotor topologies, these characteristics may significantly vary. However, based on these simplified characteristics, it can be assumed that the squirrel cage machine, with either a laminated or solid rotor, clearly displays the best electromagnetic performance. For high-speed applications, a laminated rotor cannot be considered in most cases, as the rotor would not mechanically withstand the high circumferential speeds. As an alternative, a solid rotor with a squirrel cage can be considered, International Journal of Electrical Power and Energy Systems 166 (2025) 110520 22 V. Bilek et al. Table 7 Comparison of all rotor topologies in terms of selected characteristics, based on the literature review presented. Fig. 35. Comparison of calculated normalized torque (a) and normalized phase current RMS value (b) vs. slip characteristics for 5 fundamental rotor topologies of an induction machine. The normalized values in the figures are related to the rated torque and rated current of the laminated induction machine with squirrel cage. which has similar electromagnetic parameters and significantly better mechanical properties at high circumferential speeds. The assembly of the squirrel cage in the end-ring region is particularly challenging, where it typically requires reinforcement, for instance, through sleeves, to ensure structural integrity. Despite the complicated reinforcement of the end-rings, this critical part might still not be able to withstand high circumferential speeds, as reported in [75]. Therefore, copper-coated SSR might be a suitable choice for high-speed applications, since it has excellent mechanical properties at high circumferential speeds, is easier to manufacture, and is a compromise between squirrel cage machines and simple SSR/ASSR in terms of electromagnetic properties. Generally, the main criterion for the selection of a suitable rotor topology is a compromise between the electromagnetic, thermal, and mechanical characteristics of the machine. 9. Conclusion The design of induction machines, for high-speed applications is a very complex discipline that requires a multidisciplinary analysis of electromagnetic, thermal, and mechanical aspects. Firstly, this paper sought to give a comprehensive overview of highspeed solid-rotor induction machine applications and a classification of solid-rotor topologies. For each rotor topology, its merits, drawbacks and general guidelines for proper design were highlighted. Despite all the significant differences between the individual topologies, it is not possible to select any one as the best. Secondly, this paper focused on electromagnetic modeling of HSSRIMs, concentrating on state-of-the-art techniques and their comparison, especially on analytical calculation methods and 2D/3D FEM numerical simulations. The 2D FEM numerical simulation was found International Journal of Electrical Power and Energy Systems 166 (2025) 110520 23 V. Bilek et al. to be the most efficient method for HSSRIM calculation, offering a good trade-off between accuracy of the results and computational complexity. Furthermore, several methods for advanced 2D FEM machine modeling techniques were presented that ensure greater accordance between simulated and measured data for various solid-rotor topologies. Thirdly, since thermal aspects play an important role in the modeling of high-speed machines, especially at the design stage, this paper also focused on presenting and discussing various state-of-the-art cooling and thermal modeling techniques. Significant attention was paid to analytical methods and LPTN models, as their low demands on computing power make them suitable for inclusion in multi-physics analyses. Finally, state-of-the-art techniques for simulating the mechanical behavior of rotors were described. The analyses can be divided into two categories: stress analysis, which is essential for safety assessment, and critical speed analysis, which provides information on the speed at which significant vibration occurs. Various phenomena that can be included in the calculations were described and their influence on the results discussed. The main contribution of this study is a summary of all state-ofthe-art high-speed solid-rotor induction machines and corresponding research findings, which will provide guidance for selecting a suitable solid-rotor topology and for setting up a thorough multi-physics machine design process. CRediT authorship contribution statement Vladimir Bilek: Writing – review & editing, Writing – original draft, Visualization, Conceptualization. Jan Barta: Writing – review & editing, Supervision, Conceptualization. Marek Toman: Writing – review & editing. Petr Losak: Writing – review & editing. Gerd Bramerdorfer: Writing – review & editing. Declaration of competing interest The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Vladimir Bilek reports financial support was provided by Linz Center of Mechatronics GmbH. Vladimir Bilek reports financial support was provided by Centre for Research and Utilization of Renewable Energy. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This research work has been carried out in the Centre for Research and Utilization of Renewable Energy (CVVOZE). Authors gratefully acknowledge financial support from the Ministry of Education, Youth and Sports of the Czech Republic under institutional support and BUT specific research programme (project No. FEKT-S-23-8430). Furthermore, this work has been supported by the COMET-K2 ‘‘Center for Symbiotic Mechatronics’’ of the Linz Center of Mechatronics (LCM) funded by the Austrian federal government and the federal state of Upper Austria. Data availability No data was used for the research described in the article. References [1] Pyrhönen J, Jokinen T, Hrabovcova V. Design of rotating electrical machines. 2nd ed.. Wiley; 2013, p. 612. [2] Industrial gearbox - global market trajectory & analytics. 2020. [3] Gerada D, Mebarki A, Brown NL, Gerada C, Cavagnino A, Boglietti A. High-speed electrical machines: Technologies, trends, and developments. IEEE Trans Ind Electron 2014;61(6):2946–59. http://dx.doi.org/10.1109/TIE.2013. 2286777. [4] Boglietti A, Gerada C, Cavagnino A. High-speed electrical machines and drives [special section intro.]. IEEE Trans Ind Electron 2014;61(6):2943–5. http://dx. doi.org/10.1109/TIE.2013.2286778. [5] Fernando N, Vakil G, Arumugam P, Amankwah E, Gerada C, Bozhko S. 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