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Highspeed electrical machines have been extensively researched in the last few decades for industrial and domestic applications, including compressors, vacuum pumps, turbine generators, machine tools, flywheel energy storages, and so on. Compared with lowspeed and moderatespeed conventional electrical machines, highspeed electrical machines offer advantages such as highpower density, small size, and light weight. More importantly, highspeed electrical machines can be directly connected to highspeed loads, and conventional gear boxes are no longer needed, which avoids complex gear box systems, improves system efficiency and reliability, and reduces system vibration, noise, and cost. With the evolution in the field of power electronics converters, the problems of high frequency supplies, required for highspeed operation, is no longer a restriction. The development of highspeed electrical machines is also supported by the development of highspeed bearing systems with high robustness, fewer losses, and longer lifetime.
Advantages  Disadvantages  

Singlephase PM machine 


Threephase PM machine 


In this section, the stator structures and winding configurations of HSPM machines are overviewed. The stator structures can be classified into slotted and slotless topologies, and the winding configurations can be divided into overlapping and nonoverlapping layouts. The outline is illustrated in Figure 7.
Figure 7. Outline of stator structures and winding configurations.
In general, the stator structures of slotted HSPM machines can be separated into multislot (>6slot) and minimalslot (6slot; 3slot) [27], Figure 8. The multislot stator structure is widely used for high power (>10 kW) requirements.
Reference [30] designs a 48slot/8pole HSPM machine with a rating of 2 MW at 20 krpm by multiphysics and multiobjective optimization. The optimal goal is the maximum torque per loss per mass, with a tradeoff between power density and efficiency. For aircraft turbogenerator applications, the 36slot/4pole machine topology is employed for two HSPM machines, and their rated powers and speeds are 150 kW at 17 krpm [32] and 100 kW at 60 krpm [33]. Ref. [32] mainly analyses the various loss components, and [33] focuses on thermal analysis by the finite element method (FEM). With toroidal windings, a 36slot/2pole HSPM machine [34] and three 24slot/2pole HSPM machines are designed and analyzed, including various loss components, mechanical stress, temperature distributions, and rotor dynamics [35,36,37]. In [38,39,40,41], 18slot/2pole HSPM machines are researched for different applications, such as electric vehicles and electricturbo compounding systems. In addition, 12slot/2pole HSPM machines are employed in industrial applications [42,43,44,45]. For centrifugal turbocompressors, a 50 kW, 70 krpm 12slot/2pole HSPM machine is designed and analyzed by an analytical approach [42]. Ref. [43] investigates the influence of sleeve thickness on the rotor eddy current loss, vonMises stress, unbalance vibration response, and critical speed in a 15 kW, 120 krpm 12slot/2pole HSPM machine for air blower cooling fuel cells. Then, [44] proposes a 3 kW 100 krpm 12slot/2pole HSPM machine for electric turbochargers, and the influence of the rotor length/diameter (L/D) ratio on the response time is analyzed.
Figure 8. HSPM machines with multislot and minimalslot stator structures. (a) Multislot (12slot). (b) Minimalslot (6slot). (c) Minimalslot (3slot).
Overall, for various applications and requirements, the slot number should be optimized for different optimization objectives in the design of highspeed HSPM machines. [46] investigates the influence of slot/pole number combination on the performances in highpower HSPM machines, including 12/24/27/36slot and 2/4pole. The results show that compared with the machine with a 2pole rotor, the machine with a 4pole rotor has larger stator iron loss but smaller copper loss due to shorter endwinding length. In addition, the 4pole rotor can reduce the machine axial length and improve the rotor dynamic characteristics. With the 4pole rotor structure, the influence of slot number on the cogging torque and PM eddy current loss is investigated, and the 27slot is selected due to the lowest cogging torque and small PM eddy current loss.
For lowpower HSPM machines, the minimalslot stator structure is employed to simplify the winding process and avoid the physical limitations of the small size. In the literature, three typical slot/pole number combinations are selected in lowpower smallsize HSPM machines, i.e., 6slot/4pole, 6slot/2pole, and 3slot/2pole, Figure 9, which will be reviewed as follows.
6slot/4pole
As mentioned before, the 4pole rotor is adopted to reduce the endwinding length, copper loss, and stator yoke thickness in several multislot HSPM machines [46,47,48]. With the same design considerations, a 4pole rotor is selected for 6slot HSPM machines [49,50,51,52]. Ref. [49] designs a 3 kW 80 krpm 6slot/4pole HSPM machine with concentrated windings for generators. The tooth shoe shape is optimized to minimize the eddy current loss in the rotor sleeve and increase the efficiency. Ref. [51] investigates the optimal split ratio of 6slot/4pole HSPM machines considering stator iron loss and mechanical constraints. In [52], the 6slot/4pole and 6slot/2pole HSPM machines with a rating of 2.5 kW at 100 krpm are compared to enhance output power density. With the same loss density, cooling capability, packing factor, and machine axial length (including endwinding), the 4pole machine can increase the output power density by 1.5 whilst at the same time decreasing the machine size by 33% (stator outer diameter), when compared with the 2pole machine. However, this research neglects the switching loss in the power electronics converter, which has a close relationship with fundamental frequency, this is doubled for the 4 pole machine and should be considered.
Figure 9. Lowpower smallsize HSPM machine topologies with minimalslot stator structures. (a) 6slot/4pole. (b) 6slot/2pole with straight teeth. (c) 6slot/2pole with semiclosed. (d) 3slot/2pole.
6slot/2pole
For lowpower, smallsize HSPM machines, the frequency of the 4pole machine is twice that of the 2pole machine, which leads to larger stator iron losses and switching losses in the power electronic converters. More importantly, for ultrahighspeed operation, 2pole magnets can be designed as a magnet ring or a solid PM, which have high mechanical strength and simple manufacturing processes. Therefore, the 2pole rotor is widely employed in lowpower, smallsize HSPM machines [24,27,5366].
Ref. [27] analyses several design features of highspeed PM machines for microturbines, including stator structures, winding configurations, rotor constructions, and bearing systems. It shows that compared with the multislot structure, the 6slot/2pole machine with fullpitched overlapping windings has the largest winding factor, i.e., ‘1’. [24] designs a 5 kW 240 krpm 6slot/2pole ultrahighspeed PM machine with concentrated windings. The Inconel718 sleeve is employed to protect the rotor, and the magnet rings are magnetized radially. Although the winding factor decreases from ‘1’ to ‘0.5’ due to the concentrated winding, the reduced endwinding axial length leads to the decreased machine axial length and improved rotor dynamic characteristic. In [53], for turbocharger applications, the 6slot/2pole and 3slot/2pole number combinations are compared for a 2 kW HSPM machine with a rated speed of 120 krpm and the maximum speed of 220 krpm. According to the loss analysis, the 6slot machine with significantly smaller overall machine losses is more suitable for highspeed applications. Therefore, [55,56] design a 6slot/2pole 1.5 kW 150 krpm HSPM machine with parallel magnetized magnets for automotive superchargers; ‘b’shaped easily manufactured copperbar windings are employed. [57] analyses the eddy current losses in copperbar windings due to proximity effect caused by highspeed operation and reduce the AC copper losses by different toothtip shapes and segmentation of the conductors.
In [62], for compressor applications, a 3 kW, 80 krpm 6slot/2pole HSPM machine is optimized for the highest efficiency by FEM. Then, for a 15 kW 150 krpm electrically assisted turbocharger, [63] designs two highspeed PM machine topologies: a 6slot/2pole machine with straight teeth, Figure 9a, and a 6slot/2pole machine with semiclosed slots, Figure 9b. It shows that the 6slot/2pole slotted machines have a good compromise between efficiency and torque. More importantly, in terms of rotor dynamic conditions, the 6slot/2pole machine with semiclosed slots is the better choice due to the shorter machine axial length. Researchers in [61] designed two 6slot/2pole HSPM machines with a rating of 11 kW at 31.2 krpm for gas blower applications and 3.5 kW at 45 krpm for microorganic Rankine cycle power plants. A multidisciplinary design process is proposed, and it indicates that considering the constraints of electromagnetic, mechanical, thermal, and dynamic, the design of highspeed machines cannot have a single optimum solution but can have different solutions for different requirements.
3slot/2pole
3slot/2pole number combination is the simplest structure for threephase slotted highspeed PM machines. [26] designs a 3slot/2pole 10 W 150 krpm HSPM machine for a precision handpiece (or handtool) and analyses several challenges and constraints of highspeed operation. [21] designs a 1.3 kW 20 krpm HSPM machine for a friction welding unit, and the stator iron loss is analyzed under different maximum stator iron flux densities and lamination materials. In [67], considering the sensorless operation based on the detection of the zerocrossing of the backEMF waveform, the design criteria of a 1.26 kW 120 krpm 3slot/2pole HSPM machine is investigated. It shows that under the same overall size (including endwinding) and the similar efficiency, a machine with a lower stator iron flux density has smaller inductances and lower diode conduction angles and is therefore more suitable for highspeed sensorless operation. In [68], a 3slot/2pole HSPM machine is optimized by FEM for maximum efficiency considering stator iron loss. [69] analyses and designs two 1.0 kW, 2pole HSPM machines with a speed range of 20–40 krpm. One is a 3slot/2pole slotted HSPM machine, and the other is a 2pole slotless HSPM machine. In [1], based on the 3slot/2pole slotted and 2pole slotless machines, the potentials and limits of lowpower HSPM machines are discussed in terms of electromagnetic, mechanical, and thermal aspects.
However, compared with the 6slot/2pole, the 3slot/2pole machines have inherent UMF due to the diametrically asymmetric stator structure. In [70], the influence of machine dimensions on noload UMF is investigated, and a method of reducing noload UMF is proposed, this adds extra notches in the middle of each stator tooth, . Then, in [71,72], the position and size of extra notches are optimized to reduce the noload and onload UMFs, Figure 10b, and rotor eddy current losses of 3slot/2pole PM machines, respectively, Figure 10c. They show that the optimized extra notches almost eliminate the rated onload UMF and significantly reduce the rotor losses. However, the elimination and reduction highly depend on load conditions, and the output torque is decreased slightly.
The rotor eccentricity due to manufacturing tolerances not only significantly increases the UMF but also affects the electromagnetic performances of the 6slot/2pole and 3slot/2pole PM machines, including opencircuit airgap field, backEMF, cogging torque, electromagnetic torque, etc. [73] investigates the influence of static and dynamic rotor eccentricities on the electromagnetic performance of 3slot/2pole PM machines considering eccentricity ratio, eccentricity angle, and rotor initial angle. It shows that static rotor eccentricity leads to unbalanced magnitudes and phase angles of fundamental backEMFs of the three phases but does not change the harmonic content. For dynamic rotor eccentricity, three phase backEMFs remain balanced, but exhibit asymmetric positive and negative halfperiods in the phase backEMF waveforms depending on the eccentricity angle and the rotor initial angle. The largest asymmetric backEMF waveform occurs when the angle difference between the eccentricity angle and the rotor initial angle is 90 electrical degrees. Compared with static rotor eccentricity, the dynamic rotor eccentricity has almost the same average torque and UMF but larger torque ripple and peak cogging torque. Two kinds of manufacturing tolerances, stator core gap and misalignment due to modular design also affect the electromagnetic performance of 6slot/2pole HSPM machines with toroidal windings [74], especially for the fundamental magnitudes and phase angles of the threephase backEMFs, cogging torque, and torque ripple.
Optimal Design of Slotted Machines
As mentioned before, there is no single optimum design with multidisciplinary designs and thus this section mainly shows the optimal design in terms of electromagnetic performance.
For slotted stator structure, the split ratio, i.e., the ratio of stator inner diameter to outer diameter, is an important factor for machine design since it has a close relationship with the torque capability and efficiency [21,26,75]. For the maximum torque, the optimal split ratio depends on the remanence of the magnet (Br) and the maximum stator iron flux density (B_{max}) [21]. For the maximum efficiency considering the stator iron loss, the optimal split ratio depends on the stator active length, output torque, and Bmax [68]. This also shows that the stator iron loss considered decreases the optimal split ratio. Researchers from [69] focus on two design variables for the maximum torque, the stator inner diameter and the open circuit maximum stator iron flux density. With a fixed stator outer diameter, the first design variable is equivalent to the split ratio. In addition, the stator thermal limitation is introduced to avoid the thermal issues of the stator, which means that the sum of stator copper loss and iron loss is restricted. Considering the stator iron loss and mechanical constraints, [51] investigates the optimal split ratio of a 6slot/4pole HSPM machine for maximum torque considering the influence of airgap length and rotor pole pairs. The results show that the mechanical constraints considered decrease the optimal split ratio. The limitations of stator total loss and rotor eddy current losses are taken into account in [65] since the rotor losses in smallsize HSPM machines are difficult to reduce, and the cooling capability of the rotor is relatively poor. In [66], three 6slot/2pole HSPM machines with 1, 2, and 3 slotpitch windings are optimized by FEM considering the influence of armature reaction on the stator iron loss. The stator iron loss increases with the increase of coilpitch due to the increased magnitudes of stator magnetomotive force (MMF) spatial harmonics.
For the minimum torque ripple and maximum efficiency, considering the stator iron loss, copper loss, and rotor eddy current losses, a ‘Tanguchi’ method is used in [76]. The ‘Tanguchi’ method is a simplified version of the global optimization, and different design variables with different suitable variation ranges are adopted. This method can avoid the calculation of maximum stator iron flux density.
Figure 10. Extra notches for reducing noload UMF, onload UMF, and rotor eddy current loss. (a) Extra notches for reducing noload UMF [70]. (b) Extra notches for reducing onload UMF [71]. (c) Extra notches for reducing eddy current loss [72].
The slotless stator structures with large airgaps and flux leakages have relatively small output torque, and are rarely used for conventional lowspeed and moderatespeed PM machines. For highpower, largesize HSPM machines, although the high speed can increase the power, the critical speed is limited by the rotor mechanical strength and dynamic characteristics. Therefore, a relatively large torque is required for these machines; therefore, the slotless stator structure is not a suitable choice. For lowpower, smallsize HSPM machines, the critical speed is higher than that of highpower, largesize HSPM machines due to smaller rotor diameter. Therefore, the slotless stator structures with relatively small torque can be accepted in lowpower, smallsize HSPM machines, Figure 11.
Figure 11. Slotless highspeed PM machines. (a) Airgap windings. (b) Toroidal windings.
Ref. [77] proposes a 2pole slotless PM machine for speeds in excess of 100 krpm, and it is compared with a slotted HSPM machine with the same external size. The analysis shows that the slotless stator structure with a simple construction is extremely attractive, especially for ultrahighspeed operation. Toroidal windings are employed in this slotless machine. In [27], multislot, minimalslot, and slotless machines are discussed. It shows that compared with the slotted machine, the slotless machine has a smaller opencircuit airgap flux density due to larger flux leakage and lower rotor loss due to the uniform air gap. To compensate for the reduction of flux density, the stator active length of the slotless machine is relatively long.
The comparison in [27,77] mainly focuses on electromagnetic performance. However, in [1,69], the comparison of a 3slot/2pole slotted HSPM machine and a 2pole slotless HSPM machine includes electromagnetic performance, temperature distribution, mechanical stress, and PM demagnetization. The results show that for slotless machines, the effect of armature reaction is restricted, and the PM demagnetization is avoided due to the reduced rotor eddy current loss. [50] compares a 6slot/4pole HSPM machine and a 2pole slotless machine for the reduction of noise emissions. It shows that the slotless machine has 38% smaller opencircuit power loss and 12 dB lower noise due to significantly less armature reaction.
In the literature, it can be seen that the slotless stator structure is popular for ultrahighspeed applications due to its simple structure and no teeth [50]. Example applications are: a 1200 krpm microsize machine [31], a 1000 krpm starter for a gas turbine [78], a 500 krpm mesoscale gas turbine [28,79], a 400 krpm centrifugal compressor [80,81]. Most importantly, the slotless stator may be the only solution for microscale HSPM machines. [82] presents a millimeterscale 2pole slotless bearingless HSPM machine with a rated speed of 160 krpm for electrical drive systems. The measured overall machine losses at 160 krpm are below 1 W and the measured temperature is below 45 °C.
2pole slotless ultrahighspeed PM machines are designed and optimized for minimum power losses by analytical methods in [79,80], and [83]. However, the constraints used are different, namely, the maximum stator iron flux density in [79], the maximum allowed current density in [80], and the dimensions of the rotor in [83].
In this section, the applications, advantages, and disadvantages of slotted and slotless stator structures are summarised in Table 2. The selection of slotted and slotless stator structures usually depends on the design requirements. The slotted stator structure is suitable for the requirement of large output torque, and the slotless stator structure is a better choice for the requirements of low rotor loss and microsize machine dimension.
Table 2. Applications, advantages, and disadvantages of slotted and slotless stator structures.

Slotted 
Slotless 
Application 


Advantages 


Disadvantages 


With slotted and slotless stator structures, various winding configurations can be combined for different applications. In general, the winding configuration can be divided into overlapping and nonoverlapping windings, which have a close relationship with the winding factor, endwinding length, and machine axial length.
For machines with integral, overlapping windings include fullpitched and shortpitched versions according to the coil span angle (θ_{s}), i.e., θ_{s} = π elec. deg. and θ_{s} ˂ π elec. deg., respectively. When the coil span angle is equal to the pole pitch, i.e., a fullpitched winding, the winding factor is ‘1’, and when the coil span angle is less than the pole pitch, i.e., a shortpitched winding, the winding factor is less than ‘1’.
Slotted Machine
In highpower multislot (˃6slot) HSPM machines, fullpitched overlapping windings are widely employed, Figure 12a, [32,33,38,39,40,42,43,44,84,85,86,87]. Although the fullpitched overlapping winding has a long endwinding and axial length, the output torque is usually large since the winding factor is ‘1’.
In [88], a 36slot/6pole 30 kW 20 krpm highspeed IPM machine employs a 5/6 shortpitched overlapping winding to reduce the space harmonic components of MMF and minimize the rotor eddy current losses. However, the fundamental component and the output torque are slightly decreased due to the reduced winding factor, from 0.966 to 0.933. Researchers in [43,77] also employ 5/6 shortpitched windings for a 12slot/2pole HSPM machine to reduce the rotor loss, Figure 12b. [89] employs 4 slotpitch windings, i.e., 4/12 shortpitched windings, for a 24slot/2pole HSPM machine to short the endwinding axial length and machine axial length, whose winding factor is only ‘0.48’.
For lowpower minimalslot HSPM machines, the fullpitched overlapping winding can only be used in 6slot machines. However, the 6slot/2pole lowpower smallsize HSPM machine is sensitive to the endwinding axial length, and the fullpitched overlapping winding leads to long axial lengths, low torque density, and low rotor mechanical natural frequency. In [90], the 3slot/2pole HSPM machine with concentrated nonoverlapping windings and two 6slot/2pole HSPM machines with fullpitched overlapping and concentrated nonoverlapping windings are compared for the rotor eddy current loss. With sinewave currents and unmagnetized magnets, i.e., only considering stator MMF space harmonics, the 3slot/2pole machine has the largest rotor eddy current loss, and two 6slot/2pole machines have almost the same rotor loss.
For 6slot/2pole HSPM machines, 3 slotpitch (fullpitched) windings [27,90] and 1 slotpitch (concentrated) windings [24,53,91] have been employed for highspeed applications, Figure 13a,d. [66] compares three 6slot/2pole HSPM with 1, 2, and 3 slotpitch windings and the results indicate that 2 slotpitch (shortpitched) windings, Figure 13b, are attractive for improving the power density due to relatively short endwinding axial length and relatively large winding factor, which is suitable for highspeed applications. In addition, the influence of endwinding on the torque and torque density with different stator active lengths is investigated. It shows that for relatively long stator active length, the machine with 2 slotpitch windings has the largest torque, as well as the highest torque density. Furthermore, [92] designs two 6slot/2pole HSPM machines with different layouts of 2 slotpitch windings, i.e., Machine A and Machine B, Figure 13b,c. It shows that, compared with Machine A, Machine B is an attractive machine design due to shorter axial length of endwinding, higher torque density, and smaller phase inductance. In addition, a conventional 3slot/2pole HSPM machine with nonoverlapping windings is compared with the 6slot/2pole HSPM machine with 2 slotpitch windings since they have the same winding factor, i.e., 0.866. The results show that compared with the 3slot/2pole HSPM machine, the 6slot/2pole HSPM machine with 2 slotpitch windings has advantages of high torque, low rotor loss, no UMF, and small phase inductance, which are desirable for highspeed operation.
Figure 12. Fullpitched and shortpitched overlapping windings for 12slot/2pole HSPM machines. (a) Fullpitched. (b) 5/6 shortpitched.
Figure 13. 6slot/2pole HSPMmachines with 1, 2, and 3 slotpitch windings. (a) 1 slotpitch winding. (b) 2 slotpitch winding, MA. (c) 2 slotpitch winding, MB. (d) 3 slotpitch winding.
Slotless Machine
For slotless HSPM machines, the overlapping winding is widely employed since the large winding factor can compensate for the low flux linkage caused by large airgap, Figure 14a, [1,69,81,93,94,95,96].
Figure 14. 2pole slotless HSPM machines with fullpitched and shortpitched windings. (a) Fullpitched. (b) Shortpitched.
Ref. [93] compares the overlapping and nonoverlapping windings for slotless PM brushless machines. It shows that the conventional overlapping windings can offer large power density and are more suitable for medium and large machines where the power density is an important criterion. However, it also points out that the overlapping windings may be difficult to wind, and their relatively long endwinding may not be acceptable for highspeed applications. Researchers in [94] design a 2pole slotless HSPM machine for handtool applications. With fullpitched overlapping windings, the axial leakage is analyzed by a 2D analytical model and a 3D FE model. It shows that at the axial ends of the machine, the reduction of flux density is substantial, but that the reduction in backEMF is minor due to the machine dimensions for handtool applications. In [79], a 100 W 500 krpm slotless HSPM machine adopts fullpitched overlapping windings, and its winding factor (k_{w}) is calculated by
Compared with a slotted machine with fullpitched overlapping windings, for which k_{w} = 1, the slotless machine with fullpitched overlapping windings has a relatively lower winding factor. However, although the fullpitched overlapping winding has a large winding factor, the long endwinding length and axial length may decrease the torque density, increase the copper loss, and reduce the rotor mechanical natural frequency. Therefore, the shortpitched overlapping winding with a relatively large winding factor and relatively short endwinding axial length is attractive for improving torque density. In [29,97], a 2pole slotless ultrahighspeed PM machine with a rating of 2 kW at 200 krpm is designed for cryogenic compressor applications. The machine employs 15/18 shortpitched overlapping windings to reduce the endwinding length and copper loss, Figure 14b.
In this section, different nonoverlapping winding configurations are discussed, including concentrated windings, toroidal windings, and skewed slotless windings.
Slotted Machine
In highpower multislot (˃6slot) HSPM machines with 2/4pole rotors, the concentrated winding results in a low winding factor. Take 2pole machines with 12/18/24slot as examples. The fundamental winding factors of three machines with fullpitched overlapping windings are 0.966, 0.960, and 0.958, respectively, while those of three machines with concentrated windings are 0.5, 0.5, and 0.5, respectively. Therefore, the concentrated nonoverlapping winding is not desirable for highpower multislot HSPM machines.
Compared with fullpitched overlapping windings, toroidal windings with the same winding factor have a shorter endwinding axial length, which improves the rotor stiffness and avoids rotor dynamic issues. More importantly, the airspace between the slots at the core back (Figure 15) can be used as cooling channels, increasing the machine cooling capability. These considerations have resulted in toroidal windings replacing overlapping windings in highpower multislot HSPM machines [35,36,37]. However, the outside conductors of toroidal windings do not produce backEMF but are a part of the endwinding length and consequently increase the copper loss. In addition, the leakage flux caused by outside conductors leads to eddy current losses in the machine frame [98], which may account for about 50% of overall losses in a particular design and decrease the machine efficiency. Although the heat dissipation in the frame is not difficult to deal with, it needs attention in the design process.
In [37,99], the windage losses and antidemagnetization of the 75 kW 60 krpm 24slot/2pole HSPM machine with toroidal windings are studied by 3D FEM, Figure 15. The predicted windage losses and rotor temperature are verified by the measured results. It also shows that the toroidal winding can reduce windage loss, which has a close relationship with the rotor axial length. [36] proposes a design method of HSPM machines considering electromagnetic performance, mechanical strength, rotor dynamic characteristics, and temperature distribution. Then, a 15 kW 30 krpm 24slot/2pole HSPM machine with toroidal windings is designed and prototyped. Researchers in [35] design a 24slot/2pole aircooled HSPM machine with toroidal windings. The electromagnetic performance is analysed by a FEM model, and the thermal properties with aircooling are analyzed by computational fluid dynamic (CFD) thermal model and LPTN model. By optimising the stator dimensions, especially the outer slot, the loss density is reduced, and the cooling capability is improved [100].
Figure 15. 24slot/2pole HSPM machine with toroidal windings [37].
Therefore, in multislot HSPM machines, toroidal windings are more suitable than concentrated nonoverlapping windings. However, both toroidal windings [27,101,102] and concentrated windings [24,54–65,91] are widely used in 6slot HSPM machines, Figure 16. [61] shows that although the fundamental winding factor of the concentrated winding is ‘0.5’, doubling the number of turns may not lead to large copper loss due to short endwinding compared with the fullpitched overlapping windings. In addition, with forcedair cooling, the outer slot not only supports the outer windings of toroidal windings but also can be the airduct to improve the thermal dissipation capability.
Figure 16. 6slot/2pole HSPM machines with concentrated and toroidal nonoverlapping windings. (a) Concentrated winding. (b) Toroidal winding.
In 3slot/2pole HSPM machines, the concentrated winding is a conventional winding configuration [1,21,26,67,68,69,76,103], while the toroidal winding is rarely mentioned. [101] analyses the electromagnetic performance of 2pole HSPM machines with toroidal windings and different slot numbers, including 3slot, 6slot, 9slot, and 12slot. It shows that the winding factor of the 3slot/2pole machine with toroidal windings is ‘0.5’, which is smaller than that of the 3slot/2pole machine with concentrated windings, ‘0.866’. In addition, the 3slot/2pole machine with toroidal windings cannot eliminate the 3rd harmonic in the flux linkage waveform caused by the associated 3rd airgap field harmonic or local magnetic saturation. This results in a 3rd harmonic component in the phase backEMF waveform. However, this will disappear in the line backEMF waveform and it has no influence on the torque. Therefore, in 3slot/2pole HSPM machines, the concentrated winding is a better solution than the toroidal winding with a smaller winding factor.
Slotless Machine
In slotless machines, three nonoverlapping winding configurations can be used [93,104], such as concentrated windings [80], toroidal windings [28,98], skewed slotless windings [105,106,107,108], Figure 17.
In [77], a multilayer basket winding, i.e., skewed slotless winding, and a ring winding, i.e., toroidal winding, are compared for a 50 W 150 krpm slotless HSPM machine. It shows that the skewed slotless winding has almost no endwindings and all conductors are in the airgap and magnetic field caused by PMs, which means the winding axial length is almost the same as the stator active length. However, this kind of winding with a relatively complex structure requires specialist equipment and a special manufacturing process. In the literature, there are three skewed slotless winding configurations, i.e., helical windings, rhombic windings, and diamond windings, Figure 18, [105,106,107,108]. Ref. [109] investigates the 3D torques and forces of two skewed slotless winding configurations, i.e., helical windings and rhombic windings, and their potential for highspeed applications is also analyzed. For skewed slotless windings, threephase coils are overlapped radially, which increases the winding thickness in the airgap, but has almost no endwinding, reducing the machine axial length and increasing the rotor mechanical natural frequency. However, since the coil is skewed by π elec. deg., the current direction is not ideal and may lead to extra undesired transverse torque and force.
Figure 17. 2pole slotless HSPM machines with different nonoverlapping windings. (a) Concentrated winding. (b) Toroidal winding. (c) Skewed slotless winding.
Figure 18. Phase A coils of three skewed slotless winding configurations [106,110]. (a) Helical. (b) Rhombic. (c) Diamond.
Ref. [110] investigates the influence of three slotless winding configurations, i.e., concentrated windings, toroidal windings, and helical windings, on the rotor eddy current loss of a 2pole slotless HSPM machine by 2D and 3D analytical and FE models. The results show that, with the same power and phase current, the machine with concentrated windings has the highest rotor eddy current loss due to the 3coil rotational asymmetric winding distribution, and the machines with helical and toroidal windings have almost the same low rotor eddy current loss. Although with higher rotor eddy current loss, the nonoverlapping windings with 3 coils per 2 poles are employed in a 2pole 400 krpm slotless HSPM machine due to their simplicity [80]. Refs. [93,104] review the winding topologies for slotless brushless permanent magnet machines and investigate the influence of winding topologies on the optimal design. They indicate that the winding configurations have small influence on the optimal ratio of magnet thickness to winding thickness for minimum copper loss, since the configuration only changes the coefficient in the calculation of copper loss. Several slotless machines with toroidal windings have been mentioned before and are widely employed in ultrahighspeed PM machines due to the simple winding process, short endwinding axial length, modular design, and improved cooling capability. Examples are 150 W at 1200 krpm in [31], 100 W at 500 krpm in [28], 150 W at 200 krpm in [98], 15 kW at 150 krpm in [63], and 160 krpm in [82].
In this section, the applications, advantages, and disadvantages of overlapping and nonoverlapping are summarised in Table 3.
As mentioned before, the overlapping winding includes fullpitched and shortpitched windings, while the nonoverlapping winding includes concentrated, toroidal, and skewed slotless windings. They have their own suitable stator structures and applications, which depend on the design requirements and machine dimensions.
Table 3. Applications, advantages, and disadvantages of overlapping and nonoverlapping windings.

Overlapping 
NonOverlapping 

Fullpitched 
Shortpitched 
Concentrated 
Toroidal 
Skewed 

Applications 
Multislot; Slotless 
Multislot; Slotless 
Multislot; Minimalslot; Slotless 
Multislot; Minimalslot; Slotless 
Slotless 
Advantages 





Disadvantages 





For highspeed applications, the PM machines mainly employ three different rotor structures, i.e., IPM, SPM, and solid PM, Figure 19. Those different rotor designs affect the electromagnetic performance, thermal aspect, mechanical strength, and dynamic characteristic of HSPM machines.
Figure 19. Highspeed PM machines with IPM, SPM, and solid PM rotor structures. (a) IPM. (b) SPM. (c) Solid PM.
In conventional PM machines, IPM rotor structure can offer reluctance torque and reduction of PM volume. The limitation is the conflict between the rotor flux leakage pattern and the stress in the IPM rotor iron bridges [25,84].
For highspeed applications, although the solid steel rotor may be used to withstand high centrifugal force, the edges of magnets and rotor slots suffer high mechanical stress, which may destroy the PMs and rotor iron core [47,84]. Therefore, the mechanical stress in highspeed IPM machines should be computed carefully. Ref. [111] proposes an IPM rotor design for a 24slot/4pole HSPM machine, which consists of a solid steel rotor, four rectangular samariumcobalt magnets, and four flux barriers to reduce the leakage flux. The optimized rotor structure can operate at 40 krpm considering mechanical constraints. Ref. [25] compares two 40 kW, 40 krpm highspeed machines with IPM and SPM rotor constructions.
For IPM rotors, the magnets are inserted into the rotor slots without an interference fit and no prestress between the rotor iron and magnets. Therefore, during highspeed operation, the outer iron bridge suffers not only its own centrifugal forces but also the force from the magnet. In addition, compared with SPM, the magnets are distributed unevenly in radial direction in the rotor, leading to local peak stress in the iron at the edges of the rotor slots. Therefore, the critical speed of the IPM machine is limited by mechanical stress and is far less than that of the SPM machine if it has a carbonfibre retaining sleeve. To reduce the PM volume and, therefore, the cost, [47] analyses one SPM rotor and two IPM rotor constructions for highspeed applications. With the same speed (15 krpm) and torque (7.5 Nm), the IPM rotor with twolayer magnets has a larger reduction of PM volume and a higher safety factor for mechanical stress compared with the IPM rotor with one layer magnet.
In [84], two 24slot/4pole HSPM machines with SPM and IPM rotor structures are compared with a rating of 140 kW at 24 krpm and the same key dimensions, Figure 20. Both of them are not conventional SPM and IPM rotors, the SPM rotor having four breadshape magnets and four interpole fillets, which are bandaged by an alloy sleeve (Inconel 718) to withstand the centrifugal force, whilst the IPM rotor has a spoketype IPM rotor with the Cshaped iron laminations and trapezoidal PMs. The results show that the SPM rotor structure has higher mechanical strength and efficiency, but the IPM rotor can offer a relatively low cost.
Figure 20. Highspeed PM machines with SPM and spoketype IPM rotor structures. (a) SPM. (b) Spoketype IPM.
In the literature, several methods have been proposed to improve the mechanical strength for highspeed applications of IPM rotor structures. In [23], a retaining shield rotor construction is proposed to improve the mechanical strength of highspeed IPM rotors. This novel rotor structure is a combination of siliconsteel sheet and stainlesssteel plate, Figure 21. The former with a small flux iron bridge can reduce the flux leakage, and the latter with large tensile yield strength can withstand the centrifugal forces. The research focuses on the influence of the axial proportions of stainlesssteel plates on the mechanical stress and electromagnetic performance, especially on the torque and rotor losses, and a tradeoff should be satisfied. Ref. [22] employs solid semimagnetic stainless steel to improve the mechanical strength of the highspeed IPM rotor rather than the combination of siliconsteel and stainlesssteel. The limitation of solid rotor design is the large rotor eddy current loss. This has led to the proposal of axially segmented and circumferentially slitted solid rotors for reducing the losses. These are optimized and compared in [22]. The results show that, with the same conditions, the circumferentially slitted rotor structure has significantly smaller eddy current loss than the axially segmented solid rotor. In addition, for a 36slot/2pole highspeed IPM machine, [112] employs 6 iron ribs between magnets to reduce the rotor maximum stress and protect the rotor core and magnets, Figure 22. Meanwhile, radial, Halbach, and parallel magnetized rotors are compared in [112], and the results show that the Halbach magnet array can not only produce the sinusoidal airgap flux density waveform but also reduce the torque ripple and roughly maintain the output torque.
Figure 21. A retaining shield rotor construction for highspeed IPM machines [23]. (a) Retaining shield rotor construction. (b) Siliconsteel sheet. (c) Stainlesssteel plate.
Figure 22. 2pole rotor structure with 6 iron ribs between magnets [112].
As mentioned before, the SPM rotor with retaining sleeves has better mechanical strength and higher critical speed than IPM. In general, the SPM rotor design for highspeed applications mainly focuses on the tradeoffs between electromagnetic performance and mechanical, thermal, and dynamic constrains.
For highspeed applications, the sleeve is the most important part and has a close relationship with machine performance. Firstly, although the increased sleeve thickness improves mechanical strength, the increased effective airgap length due to the sleeve being nonmagnetic leads to a decrease in airgap flux density and output torque. Secondly, sleeves made from nonconducting material with high mechanical strength have higher critical speed. However, their low thermal conductivity may lead to high maximum temperature concentration in the rotor and thus increase the demagnetization risk. Therefore, the sleeve thickness and sleeve material are widely researched in the rotor design for HSPM machines.
Ref. [113] describes a high tensile retaining sleeve material, Kevlar, it can increase the critical speed of rotor. [114] compares 36slot/2pole HSPM machines with different sleeve materials for a 117 kW 60 krpm micro gas turbine application, such as stainless steel, carbon fibre, copper iron alloy, and copper. The comparison focuses on the electromagnetic performance and temperature distribution. The results show that the copper iron alloy sleeve increases the flux leakage, and three nonmagnetic sleeves have the same flux linkage. More importantly, although the rotor with a carbon fibre sleeve has the smallest rotor losses, the rotor with a copper sleeve has the lowest maximum temperature due to high thermal conductivity and relatively small eddy current losses. In [25], carbon fibre and glass fibre sleeves for SPM highspeed machines are compared. It is shown that the glass fibre sleeve cannot withstand the centrifugal force when the rotor surface speed is larger than 150 m/s, so that the carbon fibre sleeve can allow higher critical speeds. With different materials of magnets and sleeves, the rotor mechanical stress is computed by an analytical model and verified by the FE model in [115]. The results show that the combination of Titanium sleeve and NdFeB magnet has high mechanical strength.
Apart from the sleeve material, the sleeve shape can also be designed to reduce the rotor eddy current loss. [116] proposes a new grooved sleeve design to reduce the rotor eddy current loss, Figure 23. Circumferential, axial, and mixed grooves are compared, and the results show that the Titanium sleeve with circumferential grooving has the lowest eddy current losses and the smallest maximum stress in the sleeve. In [117], a new sleeve design is employed to reduce the sleeve eddy current loss by slitting the sleeve to split the eddy current loops, Figure 24. However, the mechanical stress withstand is significantly reduced and should be checked carefully.
Figure 23. Sleeve design with axial and circumferential grooves [116]. (a) Axially grooved sleeve. (b) Circumferentially grooved sleeve. (c) Mixed grooved sleeve.
Figure 24. A new sleeve design for reducing eddy current loss [117].
In [15], a copper shield between the retaining sleeve and magnets is employed to potentially reduce the total rotor eddy currents by utilising the shielding effect. The results show that as the shield thickness is increased, the total rotor loss decreases due to the stronger shielding effect. In addition, according to the thermal analysis, with a copper shield, the maximum rotor temperature is reduced from 240 °C to 70 °C. However, the airgap flux density is reduced due to the increased effective airgap. In [118], the sleeve eddy current loss decreases with the increases of the copper shield thickness firstly and then remains almost the same since the thickness is beyond the skin depth of the copper shield. In addition, with and without a copper shield, the effect of sleeve axial segmentation on sleeve losses is investigated. The results show that a sleeve without segmentation and with a copper shield is the most effective solution to reduce sleeve losses. Ref. [119] employs a radial multilayer sleeve to reduce the rotor eddy current loss, which decreases the rotor losses and PM temperature under the same output performance and mechanical strength. Researchers in [120] propose a hybrid protective measure consists of Titanium alloy and carbon fibre. The Titanium alloy shield can not only improve the mechanical strength but also reduce the losses of the carbon fibre sleeve and magnets. Although large losses exist in the Titanium alloy shield, the maximum temperature of the sleeve with Titanium alloy is almost the same as that of the sleeve without Titanium alloy due to the high thermal conductivity of the Titanium alloy shield.
Ref. [46] investigates the influence of the ratio of pole arc to pole pitch, i.e., pole arc coefficient (α_{p}), on electromagnetic performance. With the increase of pole arc coefficient, the PM loss, cogging torque, and noload current increase due to the increased amplitude of the fundamental backEMF and harmonic components, which is verified by [32,121]. The increased EMF leads to a decreased stator active length. Therefore, a tradeoff should be satisfied based on the requirements.
When the pole arc coefficient is less than ‘1’, an interpole airgap exists, Figure 25a, which may lead to large local stress in the sleeve due to the uneven distribution of magnets [33]. Therefore, [120] compares different materials of interpole filler to reduce the sleeve stress and improve the rotor stiffness, Figure 25b, such as plastics, carbon fibre, and Titanium alloy. In [33], the materials of interpole filler are PMs, i.e., α_{p} = 1, air, glass fibre, and iron. The results show that the Titanium alloy and iron interpole fillers have the lowest sleeve stress. However, the Titanium alloy and iron interpole fillers will lead to the increased rotor eddy current loss due to high electrical conductivity. Therefore, [120,46] employ nonmagnetic nonconductive plastics as interpole fillers and [25,33,114,122] adopt SPM without pole gap, i.e., α_{p} = 1, for their different design requirements. In addition, the iron interpole filler, is employed by [84,123]. Ref. [122] analyses the rotor stress for a HSPM machine with segmented magnets retained by a carbon fibre sleeve, without a pole gap. It shows that magnet edging effect caused by segmentation leads to sleeve stress concentration but reduces the magnet tangential stress; thus, the number of PM segments should be optimized to avoid stress concentration.
Figure 25. Two SPM rotor structures with different pole gap materials. (a) Magnets with air pole gap. (b) Segmented magnets with pole filler.
There are two solid PM rotor structures: a solid PM with a sleeve and solid PM with a hollow shaft. The design considerations mainly focus on the mechanical stress due to ultrahighspeed operation.
The solid PM with sleeve is commonly used due to its simple rotor structure and easy assembly process, Figure 19c, especially for small size rotor [26,28,36,124]. In [27], the 2pole solid PM is used and magnetized as a whole, leading to the symmetry of mechanical strength and electromagnetic performance. In addition, the solid PM is segmented axially to reduce the rotor eddy current loss and to simplify the assembly process of the magnets, the high rotor mechanical strength and electromagnetic performance are unimpaired by the segmentation [36]. Ref. [124] compares different materials of magnets and sleeves for a 500 W 400 krpm 2pole slotless HSPM machine with a solid PM rotor. The magnet materials are NdFeB and Sm_{2}Co_{17}, and the sleeve materials include Titanium alloy, SUS304, Inconel718, and carbon fibre. The results show that the combination of Inconel718 sleeve and NdFeB (N42SH) magnet has high mechanical strength and the largest rotor dynamic safety factor and is the best. The SUS304 sleeve is the worst since it cannot meet the mechanical stress requirement.
In [29], a 2 kW 200 krpm slotless HSPM machine is designed for a centrifugal compressor drive application. This design employs a solid PM rotor structure, and the magnets are located inside a hollow shaft (Titanium), Figure 26a. This rotor structure can improve the rotor stiffness and significantly increase the first critical speed. [86] modifies the equation of natural frequency calculation considering the solid PM with hollow shaft, and the analytical prediction shows that the first natural frequency is well above the rated frequency. Ref. [125] proposes a novel solid PM rotor design with a hollow shaft, which consists of an amorphous rotor core, solid PM, and a hollow shaft, Figure 26b. It should be noticed that epoxy is employed in the gaps between PMs and rotor core to improve the rotor mechanical strength. The advantage of the novel rotor structure is that since the PM is inserted into the rotor core, the manufacture and assembly are simple. However, this rotor structure will lead to a large shaft loss, which may increase the rotor temperature and the demagnetization risk.
Figure 26. Solid PM with hollow shaft rotor structure. (a) Conventional hollow shaft [29]. (b) Novel hollow shaft [125].
In this section, the applications, advantages, and disadvantages of IPM, SPM, and solid PM rotor structures are summarised in Table 4. The selection of rotor structure depends on the constraints of mechanical stress and machine size. In general, the solid PM rotor structure has the highest mechanical strength, but the equivalent airgap is the largest. The IPM rotor structure has the lowest mechanical strength, but the cost is the lowest. The SPM with sleeve rotor structure has a better tradeoff between electromagnetic performance and mechanical stress, it is widely employed in HSPM machines.
In summary, the main types and subtypes of highspeed PM machines have been highlighted in Figure 27, and their applications, advantages, and disadvantages have been summarized in Tables 1–4.
Table 4. Applications, advantages, and disadvantages of different rotor structures.

IPM 
SPM 
Solid PM 
Application 



Advantages 



Disadvantages 



Figure 27. Main types and subtypes of highspeed PM machines.
The highspeed operation results in an increased number of parasitic effects which do not exist or are not important in the lowspeed and moderatespeed operation, such as stator iron loss, AC copper loss, rotor eddy current loss, windage loss, rotor dynamic characteristic, rotor vibration, and thermal aspects.
Compared with lowspeed and moderatespeed conventional machines, the stator iron loss becomes the dominant loss in highspeed machines due to the high frequency. Therefore, the calculation of stator iron loss should be included in the analysis and design of highspeed machines. In general, the Bertotti model [126] is employed in the analytical computation of stator iron loss. There are three losses included in the classic Bertotti model, namely, hysteresis loss (P_{h}), eddy current loss (P_{e}), and anomalous loss (P_{a}), as shown by
where k_{h}, k_{e}, and k_{a} are the hysteresis, eddy current, and additional loss coefficients, respectively. These coefficients depend on the material and are invariant with the frequency (f) and the flux density magnitude (B_{m}). However, for highspeed operation, since the magnetic flux waveform in the iron core is not exactly sinusoidal, the eddy current loss coefficient is a variable with frequency.
Ref. [127] develops a method to predict the iron loss in a PM BLDC machine under onload condition. It indicates that the onload iron losses is markedly higher than opencircuit iron losses. In addition, [128] calculates the rotational stator iron loss caused by the angle of lag between magnetic field strength (H) and flux density (B), i.e., noncircular flux density loci. In [129], the magnetic flux variation at each point of the iron core is obtained, and, the radial and tangential components of the flux density fundamental and harmonics are found by Fourier analysis. Moreover, the additional iron loss due to rotational magnetic flux in the iron core is considered. The results show that for highspeed machines, the stator iron loss is affected not only by the alternating flux effect but also by the rotational flux effect. In [100], the skin effect is considered when the frequency is above 2 kHz, and the accuracy of analytically predicted results is improved. In [39], the additional losses due to magnetic anomalies, manufacturing processes, and rotational fields are considered.
The copper losses in HSPM machines include two basic components, DC and AC copper losses. The DC copper loss component has a close relationship with output torque, and its thermal dependence is well understood. The AC copper loss is caused by the skin effect and proximity effect and should be analyzed because of the high frequency current, pulse width modulation (PWM), and large slot leakage flux in HSPM machines [130,131].
Skin effect is a tendency for alternating current to flow mostly near the outer surface of the conductor. The effect becomes more and more apparent with the increase of frequency, and thus it should be considered in highspeed PM machines. Proximity effect is a tendency for alternating current to flow in a smaller region due to the magnetic field caused by nearby conductors. In general, the skin effect can be eliminated when the conductor diameter is smaller than the skin depth under the rated frequency. [132] divides each winding turn into several parallel wires to reduce the diameter of each conductor and avoid skin effect. The proximity effect can be reduced by using Litz wire [133], which is constructed using hundreds of smalldiameter strands divided into many bundles. The strands in each bundle are twisted. [50] measures the noload power losses of the slotless HSPM machines with and without Litz wires. The results show that under the rated speed (75 krpm), the power loss with Litz wires is lower than 100 W, but the power loss without Litz wires is higher than 400 W. However, Litz wire also has drawbacks, such as the high cost and inferior thermal performance.
In addition, the proximity effect can also be affected by the disposition of the conductors. It is found that the proximity effect can be significantly reduced when the conductors are located at the bottom of slots since the additional losses are concentrated at the conductors at the top of the slot due to the influence of the magnetic field caused by the PM [131,134]. However, it is worth noting that for highspeed slotless machines, the magnetic field generated by the PM is much larger than the field caused by the nearby conductors, and therefore the proximity effect can be neglected [28].
In highspeed machines, although the rotor eddy current loss is relatively small, the poor cooling capability may result in overheating in the rotor and demagnetization of magnets. Therefore, the rotor eddy current loss should be considered and reduced. In general, two factors lead to the rotor eddy current losses, the first is the space harmonics due to the armature MMF and slot openings, and the second is the time harmonics of armature current caused by the PWM. However, in HSPM machines, the large airgap reduces the effect of space harmonics, and the time harmonics play the dominant role in generating the rotor losses [15,135].
As mentioned before, compared with 6slot/2pole HSPM machines, the 3slot/2pole HSPM machine has the largest rotor eddy current loss due to the space harmonics from the armature MMF, this loss significantly increases as the speed increases. In [110], the 2pole slotless HSPM machines with toroidal, helical, and concentrated windings and square/sinewaves generated by PWM inverters are compared in terms of the rotor eddy current loss. The results show that machines fed by squarewave PWM voltages have higher rotor eddy current losses compared to the ones fed by sinewave PWM voltage due to the larger time harmonics of armature current. In addition, slotless machines with concentrated windings have the largest rotor losses due to the space harmonics from the armature current.
In [136], the 24slot/4pole HSPM machines with Halbach and parallel magnetized rotors are compared in terms of rotor losses. According to the time and space harmonic analysis, the machine with a Halbach magnetized rotor has smaller rotor losses than the machine with a parallel magnetized rotor. However, the segmentation of the Halbach magnetized rotor is not mentioned, which may have a significant effect on the reduction of rotor losses. Ref. [137] discusses the influence of magnet thickness on the rotor eddy current loss of the 3slot/2pole HSPM machines with a constant output torque. Since the phase current and armature reaction decrease with the increase of magnet thickness, the rotor eddy current loss decreases. Ref. [72] adopts the auxiliary slots in the 3slot/2pole PM machines to reduce the rotor eddy current loss by partially cancelling the asynchronous harmonics produced by the armature field and slot, which has been mentioned in section III. A. In addition, [24] increases the stator teeth width to reduce the rotor eddy current loss.
Except for reducing the influence of space and time harmonics, splitting the eddy current loops in the rotor is a direct solution to reduce the magnet eddy current loss. Although [138] believes that segmenting the magnets is better for low speeds and [65,116] consider that PM segmentation will complicate the manufacturing process and is difficult for small size rotors, several large size HSPM machines still employ the rotor magnet segmentation to reduce the magnet eddy current loss [3,139]. In [120], the magnet is divided into three segments per pole in the circumferential direction. Axial segmentation of magnets is adopted in [30].
The windage loss results from the aerodynamic loss when the rotor rotates, and it becomes significant as the speed increases. In general, the rotor is modelled as a cylinder to find the windage loss [140], the result is given by equation 3
where ρ_{air} is the air gap density, ω is the angular speed, R_{r} and l_{a} are the rotor radius and length, respectively. C_{f} is the friction coefficient and is determined by the air gap structure and rotor surface condition. Compared with the analytical method, fluid field analysis is a more accurate method to calculate the windage loss since the friction coefficient is difficult to determine for the theoretical analysis.
Ref. [37] studies the windage loss of a 24slot/2pole HSPM machine with a rated speed of 60 krpm by the 3D FEM of the fluid field and an experimental study. It shows that the windage loss is a large part of the total losses and is larger than the core loss at the rated speed. Refs. [129,139] show that the windage losses increase with the increase of rotor speed, rotor roughness height, and ventilation speed (axial cooling air velocity). Therefore, it is advantageous to use a sleeve material with a smooth surface and employ a suitable ventilation speed to balance windage losses and cooling conditions.
During highspeed operation, the rotor has a great amount of rotational energy and a small amount of vibration energy. The purpose of rotor dynamics is to make sure the vibration energy is as small as possible. For highspeed machines, it is very important to accurately predict the natural frequency of the rotor to avoid the rated operation frequency close to the natural frequency.
In [141], FE analysis is used to establish the dynamic model of the rotor and compute its natural frequency. The results show that the shaft extension has a significant influence on the natural frequency, and in order to move rotor bending modes beyond the operating speed range, the shaft should be short and have a large diameter, i.e., small L/D ratio [44]. However, in [28], the length of the shaft is adjusted such that the rated speed (500 krpm, 8.333 kHz) falls between the second and the third bending modes. Ref. [142] investigates the influence of rotor static eccentricity on the rotor vibration by unbalanced magnetic force and eccentric mass force. The results show that the eccentric mass force leads to fundamental frequency vibration, which is the main source of rotor vibration. In addition, the static/dynamic rotor eccentricities have significant influence on backEMF, cogging torque, and UMF [73,143,144,145], which has been mentioned in Section 3.
In highspeed machines with high power density and small size, thermal analysis is necessary due to large machine loss and low cooling capability of the rotor. The computational fluid dynamics (CFD) method is adopted to calculate the temperature rise distribution, in which the coolant flow rate, the velocity, and the surface heat transfer coefficient should be determined simultaneously.
Ref. [32] optimizes the stator structure by the CFD method, and the results show that most axial coolant flows through the inner and outer slots whilst little enters the air gap. The highest winding temperature is found near the outlet, which is approximately 96.0 °C. The hottest spot occurs in the middle of the rotor, which is 125.5 °C. Since the complex modelling and meshing procedure requires certain skills and is quite timeconsuming, a timesaving lumpedparameter thermal network (LPTN) linked with the CFD modelling is employed in [35]. In this hybrid method, the fluid and temperature fields are firstly evaluated by the CFD modelling, and then the LPTN model is created.
However, in practice, both electromagnetic and thermal solutions should be done simultaneously, accounting for the inter dependence among the thermal and electromagnetic fields, mechanical and aerodynamic aspects, as well as the cooling techniques, particularly the temperature effect on the PM characteristics, the stator copper and iron losses, and the rotor PM eddy current loss etc. There remain much to be further investigated.