High-Speed Synchronous Reluctance Machines: Materials Selection and Performance Boundaries

This article presents a comprehensive comparative design exercise of synchronous reluctance (SyR) machines considering different soft magnetic materials and a wide range of speeds. First, a general design methodology able to consider all the consequences of selecting different materials is presented. In fact, magnetic nonlinearities, rotor structural limitations, and the rise of both stator and rotor iron losses are all considered. The adopted design approach allows achieving optimal stator and rotor geometries balancing all these competitive multiphysics aspects and keeping constant the cooling system capability. Both silicon–iron (SiFe) and cobalt–iron (CoFe) alloys with optimized magnetic and mechanical performance are examined to assess the maximum capabilities achievable with an SyR machine technology. The adoption of CoFe alloys leads to machines that outperform the SiFe counterparts up to a certain speed, above which, machines with SiFe provide better performance. Indeed, in the lower speed range, the effect of the higher saturation flux density of the CoFe materials is dominant, while for higher design speeds, their higher iron losses and lower yield strength, with respect to the SiFe ones, make the latter more convenient. All the design considerations are finally validated by comparing the predicted performance with the experimental test results on a 6.5-kW, 80-kr/min SyR machine prototype.

High-Speed Synchronous Reluctance Machines: T HE ever-increasing variety of soft magnetic materials in terms of chemical compositions, manufacturing processes, and thermal treatments makes the electrical machine design a challenging task [1]. The material selection is particularly important when designing high power density electrical machines where the constituent materials are pushed to the working limits in order to maximize their exploitation [2]. Indeed, transportation applications in the aerospace sector require impressively high power density levels in order to compete with the mechanical counterparts [3]. Similar challenges are faced by the automotive industry [4], although the electrical machine requirements are different.
Nonoriented cold-rolled carbon steel with different silicon content [silicon-iron (SiFe)] and alloys with different ratios of cobalt-iron (CoFe) are the main soft magnetic material families commercially available in a wide range of lamination thicknesses and thermal treatments [5]. The first one covers the majority of the market share, whereas the second is mainly used in high power density applications given its higher cost (about one order of magnitude) and higher saturation flux density [6]. Along with the latter, magnetic permeability, iron losses, and mechanical strength are the main performance indexes affecting the material selection.
Clearly, the soft magnetic material choice depends on the considered electrical machine topology and related needs. Regarding high-speed electrical machines, permanent magnet synchronous machines, induction motors, and switched reluctance topologies could benefit from the adoption of high grade or thin gauge SiFe or CoFe materials [2], [7], [8].
The iron losses at the maximum operating frequency and the saturation flux densities are the main performance factors driving the choice of the soft magnetic material to be adopted. The yield strength also plays a fundamental role when designing high-speed induction machines or interior permanent magnet synchronous with laminated rotor back iron [9], [10]. Regarding electrical machines for transportation applications, the material selection is also influenced by the maximum torque-speed profile along with the operating cycle [11]- [13].
Indeed, the driving cycle defines the material choice correctly balancing the competitive needs of having high saturation level, low iron losses, and high yield strength [14]. The soft magnetic material selection is particularly challenging when designing high-speed electrical machines featuring complex rotor structure such as synchronous reluctance (SyR) machines. The latter have been historically employed in lowspeed general purpose applications and are becoming more and more industrially accepted due to their superior performance in terms of efficiency when compared to induction motors [15], [16]. Their simple rotor construction and the capability to operate with a wide constant power speed range with the aid of low energy density magnets or small amount of rare-Earth-based magnets are further benefits justifying the automotive sector attention toward SyR machines [17].
The conflicting requirements between the rotor structural integrity at high speed and the electromagnetic and thermal performances severely influence the design process, including the soft magnetic material selection [18]- [20]. A comparative study of low-speed SyR machines with different grades of SiFe has been reported in [21] mainly assessing the efficiency aspects of the material choice. An FE-based design optimization procedure has been presented in [22] where the authors evaluate the SyR maximum power capability as a function of the speed for different soft magnetic materials. Another interesting attempt to include the structural aspects within the design workflow has been presented in [23]. A linear analytical model based on the magnetic equivalent circuits has been used to identify the maximum power capability of SyR machines as a function of the maximum design speed. Although these two studies attempt to assess the effect of the structural limitation on the electromagnetic performance, they both consider a fixed stator design and current density disregarding the thermal impacts of the increased iron losses. Indeed, if the cooling system is kept constant during the design exercise, as the speed increases, the ratio between Joule and total losses has to decrease in order to maintain the same total losses and so approximately the same thermal behavior. Otherwise, a different cooling system has to be designed, leading to unfair comparison.
This article presents a comprehensive design approach of high-speed SyR machines leading to the full definition of both stator and rotor geometries considering the limitation introduced by the rotor structural behavior and thermal constraints of the adopted materials and cooling system. The design exercise is then carried out considering different types of soft magnetic materials in order to assess the maximum power capability of this machine topology as a function of the design speed. The proposed design approach is based on the works presented in [24], improved in [25]- [27], and implemented in [28]. In particular, the design routine suitable for high-speed SyR machines presented in [26] is here extended in order to evaluate different stator and rotor soft magnetic materials. Along with the rotor structural limitations and the stator losses, also the rotor iron losses are included in the design workflow. By doing so, it is possible to fairly assess the SyR performance boundaries achievable with the considered materials. After introducing the design procedure in Section II, the soft magnetic material selection is discussed in Section III. The results of the comparative design exercise are then reported in Section IV highlighting the performance boundaries that each selected material can achieve as a function of the design speed. The optimal performance and machine geometries are then analyzed in Section V along with a thermal behavior assessment. The proposed analysis and considerations are finally experimentally validated in Section VI where several tests results are reported of a 6.5-kW-80-kr/min SyR machine designed for an electrical-assisted turbocharger [29].

II. DESIGN PROCEDURE
Considering a given outer machine envelope and cooling system capability, the performance of an SyR machine can be expressed as a function of two independent variables [24]: the split ratio (sr) and the magnetic ratio (mr). The first one is the ratio between the air-gap diameter and the external stator one, whereas the latter is the ratio between the air-gap flux density (B g ) and the iron flux density (B fe ). All the other geometrical quantities, which are reported in Fig. 1(a), can be expressed in terms of these two variables. In Section II-A, the main design equations are outlined, while in Section II-B, a computational efficient FE-based procedure to refine the analytically predicted performance is presented.

A. Analytical Design Equations
Both torque and power factor are investigated since their dependency from the design variables sr and mr is different. In fact, the torque depends both on inductances and machine capability to produce the stator MMF and flux, whereas the power factor depends only on the d-and q-axis inductances.
Indeed, the torque can be calculated as where p is the number of pole pairs, i d and i q are the d-and q-axis currents, respectively, λ d and λ q are the d-and q-axis flux linkages, which can be written as follows: where the term L dq considers the cross-coupling effects that are neglected in the first step of the design process. The d-and q-axis inductances are the sum of the magnetizing (L dm and L qm ) and leakage components where L s is the leakage inductance, equal for both axes, while L q−rib is the additional leakage inductance due to the shunted flux via the iron ribs. The power factor can be deduced from the steady-state vector diagram reported in Fig. 1(b), in which λ is the flux linkage, I m is the magnetizing current, I fe considers the iron losses, E is the electromotive force, R s and V are the stator resistance and voltage, respectively, and I is the total stator current. If the influence of the stator winding and equivalent iron losses resistances is neglected, the power factor pf coincides with the so-called internal power factor ipf, defined as the cosine of the angle between the induced voltage E and the current I m vectors.
By defining the d-axis flux per pole (2 R r L B g ), the tooth width and stator yoke radial thickness can be evaluated given a certain iron flux density B fe . Considering a uniform distribution of the equivalent rotor slots and imposing barriers having the same permeance and the total iron thickness along the q-axis equal to the stator yoke thickness allows fully defining the rotor geometry [24]. The magnetizing components of both d-and q-axis inductances and the leakage inductance (L dm , L qm , and L s ) can be then calculated as a function of the design variables sr and mr, as in [24] and [25].
The d-axis current component can be deduced from Ampere's law (when i q = 0) [26], while the q-axis component can be calculated knowing i d and the maximum current I max . The latter depends on the cooling capability k cool (in terms of W/m 2 ) and geometric parameters, as reported in (4) where k fill is the filling factor of the slot, A slots is the slots area, R s is the outer radius of the machine, ρ is the copper resistivity, and L and L ew are the axial winding and the end-winding lengths, respectively. The above maximum current calculation is based on the conservative hypothesis that Joule and iron losses (P fe ) of both stator and rotor are dissipated via the stator cooling system. The assumption of considering constant total losses determines a reduction of the torque capability as the maximum speed requirement increases. In fact, the torque derating is correlated with the reduction of copper losses in order to balance the extra iron losses at high speed. Therefore, the cooling system can be considered invariant as the speed requirement increases. Further thermal considerations will be provided in Section V-C. The iron losses are estimated using (5) where k fe is an empirical coefficient, M fe is the iron mass, k h and k e are the hysteresis and the eddy current coefficients, respectively, α and β are exponential coefficients for the frequency and the flux density, respectively, and f is the electrical frequency. The torque capability is also reduced by the rise of the structural rib thickness, which increases with the centrifugal forces affecting the rotor flux guides. The selection of the iron ribs thickness, position, and orientation along each barrier can only be accurately performed via a structural FE study. However, a simplified analytical formulation considering only the steady-state centrifugal force F c (i ) acting on the i th flux guide is usually adopted [22], [30] to estimate the total iron rib w rib (i ) of the i th barrier where k s is a safety factor, σ s is the yield strength of the rotor lamination, m fg is the sum of the flux guide masses that are sustained by the i th iron bridge, R fg is the center of gravity of the same masses, and ω m is the mechanical speed. Once the total rib thickness per each barrier is calculated, its optimal distribution along the flux barrier can be identified with a structural finite element analysis (FEA) [27]. However, wherever the iron ribs are located along the flux barrier, it can be assumed that they electromagnetically act in the same way in terms of average torque. In other words, the torque reduction due to the iron ribs mainly depends on their size and not on their distribution and position along the flux barriers. Considering n barriers per pole and supposing that the deterioration of the performance is proportional to the average value of the iron ribs w avg (7) then the rib shunted q-axis flux λ rib can be approximated as in (8), assuming that all the ribs share the same saturation level (B sat ) The increment of the q-axis inductance can be then calculated allowing to estimate the worsening of both torque and power factor due to the structural iron ribs.
The described design procedure has been applied to a case study whose details are reported in Table I. The dependencies of main performance indexes from the split and magnetic ratios are analyzed by contour plots shown in Fig. 2(a1)-(f1) for a design speed of 60 kr/min. show considerable mismatches between the analytical and the FEA results. Fig. 2(c1) shows a comparison between the estimated stator iron losses and the FE computed ones. Clearly, the former does not match with the FE computation, both in quantitative and qualitative terms. This mismatch is mainly due to the nonuniform distribution of the flux density within the machine.
The analytical and FE calculation of the d-axis inductance L dm is shown in Fig. 2(d1). It can be noticed that the saturation of the d-axis greatly reduces the d-axis inductance, leading to a worse estimation of torque and power factor. Fig. 2(e1) confirms that approximating the increment of the q-axis inductance proportional to the average total bridge is acceptable since the analytical contours clearly match the FE ones. Fig. 2(f1) shows the ratio between L dq i q and L dd i d , which quantifies the cross-coupling effect. As expected, in the design plane region where the cross coupling is more pronounced, the torque and power factor error is higher.

B. Bridging the Analytical Design Gaps
Despite the advantage of the fast performances evaluation, the pure analytical design approach shows evident accuracy limits in the performance estimation. These are mainly due to the disregarded or approximated aspects of the analytical modeling, such as the saturation of the d-axis, the crosscoupling effects, and the estimation of the iron losses. In order to properly consider all the aforementioned design aspects, a comprehensive procedure able to adjust the analytical-FE mismatch due to the saturation effects, increased iron losses, and iron ribs incurring at high speed has been proposed in [26]. Such adjustment procedure is here extended in order to include the evaluation of the rotor iron losses needed to determine the maximum current. The analytical design procedure described in Section II-A is first carried out for the whole sr-mr plane. Then, for each machine placed at the corners of the design plane, an iterative FE procedure is performed in order to accurately identify the iron losses (and so the rated current) and all the inductance components. The proposed procedure is summarized in the flowchart in Fig. 3 and consists of the following steps.
1) At the first step, the maximum current is calculated using (4) and the analytical estimation of the stator iron losses. 2) Then, a transient nonlinear FEA is carried out to calculate both stator and rotor iron losses, allowing to evaluate the mismatch with the analytical values. 3) If the relative error is less than a predefined threshold, the algorithm proceeds to the next step. Conversely, the iron losses (both the stator and rotor ones) are updated and a new maximum current is calculated. 4) Once the correct i d , i q currents are identified, a couple of linear FEAs are carried out with the permeability frozen to the values obtained from the previous nonlinear FE simulation. The first linear FEA is done supplying only with the d-axis current, while the second is done supplying only with the q-axis component. By doing so, it is possible to split both d-and q-axis inductances into two components allowing also the characterization of the cross-coupling term. A further subdivision of the inductances is also performed by calculating the magnetizing components integrating the air-gap flux density. 5) The correction factors can be then calculated as the ratio between the finite element ( FEA ) and the analytical ( AN ) prediction where x can be either the iron losses P fe , the magnetizing inductances L dm , L qm , the slot leakage inductance L s , or the cross-coupling inductance L dq of the i th corner machine. Once all the corner solutions of the design plane sr-mr are identified with the procedure summarized above, all the correction factors can be extended to the whole sr-mr plane with a linear interpolation [i.e., defining the function k x (sr, mr)]. Finally, torque and internal power factor of each machine in the design plane sr-mr can be recalculated by adjusting the inductance components and the iron losses with the respective interpolated correction factors [k x (sr, mr)].

III. MATERIALS SELECTION
The performance and optimal geometry of a high-speed SyR machine heavily depends on the selected soft magnetic material. In fact, specific iron losses p fe and yield strength σ s are intrinsic characteristics of the material affecting the decrement of the performance as the speed increases: the former reduces the maximum current, whereas the latter increases the q-axis inductance. Obviously, it is possible to achieve the best performance adopting a material with low specific iron losses and high yield strength. Unfortunately, materials with high σ s generally present high specific iron losses and vice versa. Indeed, high mechanical strength is achieved by reducing the grain size of the material, which in turn increases the iron losses [31]. Therefore, the choice of the lamination material is the result of a comparative study in which materials with different magnetic and mechanical properties are analyzed. The best solution, i.e., the one which guarantees the maximization of the output power for a defined rotor speed, will depend on the material characteristics. Fig. 4 reports a comparison in terms of yield strength, saturation flux density, and iron losses of some commercially available grades of SiFe (•, , , and ) and CoFe ( and ) under their respective trade names. The iron losses have been calculated with a modified Steinmetz model [see (5)] with the coefficients fitted with the data provided by the manufacturers [32]- [35]. The saturation flux density is considered as the point where a 5% flux density improvement corresponds to a 50% increment of the magnetic field.
Common SiFe steels (• in Fig. 4), such as M250 and M330 with 0.35-/0.5-mm lamination thickness, are usually used in medium-to-high performance volume-manufactured machines [36] due to their medium electromagnetic and mechanical characteristics. Reducing the lamination thickness (from the more common range 0.35-1 to 0.1-0.2 mm) of standard SiFe alloys (with Si content around 3%) is a commonly adopted approach to lower the eddy current losses (e.g., NO10, NO20, Arnon 7, and in Fig. 4). Reducing the lamination thickness is a less expensive way leading to higher flux densities with respect to the option of increasing the Si content even though it slightly complicates the assembly process [5]. It is well known that the silicon (Si) addition to the carbon steel increases the electric resistivity and the mechanical hardness at the cost of a lower saturation flux density and permeability [37]. The maximum Si content has been historically limited to 3.5% by manufacturing considerations since higher percentages make the alloy more brittle and hard and so difficult and expensive to produce. In the last decade, this challenge has been overcome due to the adoption of tailored manufacturing processes such as chemical vapor deposition [38] and diffusion annealing [39] (e.g., 10JNEX900 and 10JNHF600, in Fig. 4) pushing the maximum Si content up to 6.5% [40]. Extremely high yield strength (above 800 MPa) has been achieved with special production techniques (e.g., dislocation strengthening [41]), at the cost of increased iron losses (e.g., 35HXT780T, in Fig. 4).
The cobalt content makes the CoFe alloys more expensive than the SiFe contender but allows higher saturation flux densities. In general, CoFe alloys have lower yield strength with respect to SiFe (i.e., Vacoflux 48, Vacoflux 50, and Hiperco 50A, in Fig. 4). This metric can be improved by lowering the Co content (e.g., Vacoflux 17 and Hiperco 27) or by adding small percentage of vanadium and/or by additional alloying the material with niobium (Hiperco 50HS, Vacodur 49, and Vacodur S Plus, in Fig. 4), all at the cost of higher iron losses and lower saturation levels [36]. However, the most important leverage to control the tradeoff between mechanical and electromagnetic performances of CoFe alloys is the annealing process. Lower annealing temperatures decrease the grain sizes, leading to higher yield strength and lower electromagnetic properties (e.g., Hiperco 50HS, Vacodur 49 mec, and Vacodur S Plus in Fig. 4). By doing so, stators and rotors can be manufactured from the same material and then be subject to different heat treatments in order to achieve a magnetically optimized stator and a rotor with a higher yield strength. It is worth to underline that the cutting process of laminated materials worsens the electromagnetic performance [42]. However, SiFe alloys are usually used without expensive post-manufacturing thermal treatments, which would restore part of this magnetic property deterioration [43]. On the contrary, for CoFe alloys, the annealing process is indispensable to obtain the desired magnetic and mechanical characteristics [5].
The above considerations reveal the rationale behind the manufacturers' common choices of adopting SiFe alloys, with standard Si content, in low-to-medium power density applications and/or high-volume electrical machines production. On the contrary, CoFe alloys are relegated to niche applications (e.g., aerospace and motor sport) where the higher power density benefit, allowed by the higher saturation levels, outweighs the higher cost [3].
Analyzing Fig. 4, it is already possible to envisage the lack of an absolute winner, i.e., a soft magnetic material outperforming the others for every design speed. In fact, the relative vertical positions of the considered materials change as the frequency increases.
With the aim of analyzing a wide range of materials, five different steel grades have been investigated for the comparative study shown in Sections IV and V: a common SiFe grade (M250-35A), a high-performance SiFe having low iron losses and good mechanical performance (10JNHF600), SiFe alloys with outstanding yield strength (35HXT780T), and two CoFe alloys (Hiperco 50A and Hiperco 50HS). Table II summarizes the stator-rotor sets of materials considered: three combinations feature the same materials for both stator and rotor, while the other two have materials optimized magnetically for the stator and mechanically for the rotor.

IV. PERFORMANCE BOUNDARIES
The design procedure described in Section II is applied in the following for the machine parameters reported in Table I. Several maximum speed requirements (from 0 to 120 kr/min) and the material combinations listed in Table II are explored. In order to maximize the magnetic exploitation of each material, all the machines have been designed considering iron flux densities (stator and rotor yokes and stator teeth) proportional to the flux density at the knee of the respective B-H curve. In particular, the stator and rotor yokes flux densities have been set equal to the knee flux density, whereas the stator tooth one is assumed to be 20% higher. Fig. 5(a) reports the torque of the optimal designs in the sr-mr planes for each design speed and for all the considered soft magnetic materials. Fig. 5(b) shows the power factor of the same designs. It is worth to underline that the optimal design for a given speed and combination magnetic materials is hereafter considered as the one providing the maximum torque in the plane sr-mr. Fig. 2(a2) reports the location of the maximum torque design with a marker. Clearly, another design solution in the plane sr-mr can be chosen (e.g., the one providing the maximum internal power factor). However, the maximum torque selection criterion has been adopted because the aim of this comparative design exercise is to assess the performance boundaries of SyR machine topology in terms of output power as a function of the speed and magnetic materials. Fig. 5(a) and (b) shows the good match between the performance estimated with the proposed method and the FEA-calculated ones. As expected, torque and power factor decrease with the speed regardless of the alloy type.
All SiFe materials combinations start from the same torque values, but the standard one (M1) deteriorates with a higher rate with respect to the other two (M2 and M3). On the contrary, the power factor of the optimal machines featuring the standard SiFe (M1) has a similar trend to the one achieved with the high-grade SiFe (M2). Adopting SiFe alloy with optimized mechanical property on the rotor (M3) allows to improve both performance indexes but mostly the power factor. These improvements are even more evident for higher speeds.
CoFe steels outperform the SiFe contenders only up to a certain speed. It is worth to underline that the CoFe machines deteriorate more quickly than the SiFe ones both torque and power factor-wise. Indeed, considering machines at 1 and 80 kr/min, M2 (M3) torque worsens about 38% (34%), while M4 (M5) torque worsens about 66% (61%). Similarly, the power factor decrements are about 26% and 17% for M2 and M3 and circa 33% and 22% for M4 and M5, respectively. The adoption of alloys with higher saturation levels, e.g., CoFe, leads to a higher q-axis flux leakage due to the structural ribs with respect to the SiFe cases. This is the main cause of the bigger rate of performance decrement of CoFe alloys with respect to the SiFe ones. Fig. 5(c) reports the maximum output power as a function of the speed for all materials. Each considered material combination features a threshold speed above which it is not convenient to further increase the speed as a mean to push the output power and so the volumetric power density for a given cooling system. This threshold speed is clearly higher when adopting high-performance SiFe alloys. For M1, this speed limit is 50 kr/min, and for M2 and M3, it is about 90 kr/min, while M4 and M5 reach 50 and 60 kr/min, respectively. The maximum output power is also approximately 10% higher when using high-performance SiFe alloys (e.g., M2: 6 kW and M4: 5.5 kW) with respect to the CoFe ones. The adoption of standard SiFe allows reaching a maximum power 30% smaller than the more performing high-grade SiFe M2. As previously stated, CoFe alloys outperform the SiFe ones up to a certain speed, circa 55 or 65 kr/min (considering M4 and M5, respectively). The power gain is maximum around 40 kr/min and it is about 17%. Above this speed, the power improvement decreases until it becomes negative around 55 and 65 kr/min after which SiFe alloys are definitely more convenient. Adopting alloys with optimized mechanical performance for the rotor (e.g., M3 and M5) allows to improve the output power of almost 10% for both SiFe and CoFe alloys. This improvement is more evident at high-speed levels where the iron ribs dimensions become significant and have a major effect on the overall performance.

V. ANALYSIS OF THE OPTIMAL DESIGNS
The comparative design exercise, reported in Section IV, shows that the maximum speed requirement is crucial for the selection of soft magnetic materials.
The optimal machine geometries change as the speed increases and vary according to the selected soft magnetic materials. In fact, the location of the maximum torque design in the plane sr-mr [highlighted with the marker in Fig. 5(a2)] changes with the speed and consequently also the optimal geometry changes. Fig. 6(a) and (b) reports the split and the magnetic ratios of the torque-wise optimal machines as a function of the speed. Their global trends are not affected by the chosen lamination material. In fact, both design variables decrease as the speed increases, but the rate of decrement depends on the materials.
In Section V-A, the optimal geometries are analyzed in detail, while in Section V-B, the rationale behind these trends is investigated separately evaluating the effects of the iron losses and iron ribs increments. A thermal assessment of the selected machines is finally reported in Section V-C.

A. Optimal Geometries
In Fig. 7, the cross sections of the optimal machines at four different speeds (20,40,80, and 120 kr/min) and their flux density distribution at the rated condition are reported. As the speed increases, the rotor radius always decreases, but machines featuring rotor materials with low yield strength (e.g., M2 and M4) present smaller rotors with respect to M3 and M5. Similarly, the decrement of the optimal magnetic ratio implies the reduction of the stator tooth (w t ) and yoke (l y ) as well as the rotor flux guide dimensions, as reported in Table III. This decrement is again less pronounced when adopting a rotor lamination material with optimized mechanical performance. It is worth to underline that the selected Cross section of the optimal machines obtained for different lamination materials. optimal solutions for certain speeds and material combinations are definitely unconventional given the low split ratio and the high tooth length. These particular geometries could lead to manufacturing challenge, but from the electromagnetic point of view, they provide the maximum torque. Selecting another solution in the sr-mr plane, e.g., geometries with conventional proportions (and so smaller tooth length), would surely be suboptimal given their lower torque production capability.
Table III also reports the average torque, the iron losses, the torque reduction due to the iron ribs, and the maximum current reduction due to the iron losses. For each material combination, both percentage variations ( T rib and I max ) are calculated in relation to the respective values of the low-speed designs (1000 r/min).
The iron losses in CoFe machines are always higher compared to the high-performance SiFe ones (e.g., M2 and M3). At high speed, 80 kr/min, this aspect heavily affects the average torque, via the reduction of the maximum current I max [see (4)]. However, at 40 kr/min, the higher knee flux density of the CoFe steel compensates for the reduction of the stator current due to the iron losses rise, and thus, the torque is higher with respect to the corresponding SiFe machines.
It is also worth noticing that although the CoFe alloys have lower yield strengths, the torque reductions due to the iron ribs ( T rib ) are smaller with respect to the SiFe ones. This is mainly because the optimal CoFe machines have a lower rotor radius.

B. Behind the Boundaries
In this section, the trends shown in Fig. 6 and so the optimal geometries are justified analyzing the constant torque  and power factor loci in the design plane sr-mr for three different rotor speeds separately considering the structural and iron losses limiting factors. For the sake of brevity, only the most performing material combination above 65 kr/min (i.e., M3) is shown and discussed as the others behave in a similar manner. Fig. 8(a)-(c) shows the constant torque loci calculated considering only the effect of the iron losses for three different speeds (1, 60, and 120 kr/min). As the speed increases, the maximum torque designs ( at 1 kr/min, • at 60 kr/min, and at 120 kr/min) move toward the bottom-right side of the sr-mr plane, and therefore, the optimum split ratio increases, whereas magnetic ratio decreases. This trend can be explained considering that the maximum torque location is a compromise between the needs of maximizing the machine anisotropy [L d /L q , Fig. 9(a)], the magnetizing current i d [ Fig. 9(b)], and the q-axis current i q [ Fig. 9(c)]. For sake of brevity, Fig. 9 reports the contours only at 60 kr/min together with the maximum torque locations for all the three speeds (1, 60, and 120 kr/min). Indeed, L d /L q and i d are speed independent when the iron ribs are neglected, while the q-axis current i q is affected by the iron losses and so by the speed. As the speed increases, the higher iron losses lead to a q-axis current reduction (although the contour shape of i q is not affected). Therefore, the maximum torque design moves toward the area of the design plane with lower stator and rotor iron losses [reported in Fig. 9(d) and (e)] and higher magnetic anisotropy [see Fig. 8(a)]. Along the same direction, the internal power factor increases as shown in Fig. 9(f) because it follows the saliency ratio improvement. Fig. 10(a) and (b) reports the constant torque loci obtained considering only the iron ribs effect for two different speeds (60 and 120 kr/min). As the speed increases, the maximum torque designs move toward the left because the lower the split ratio the lower the average total bridge w avg , as reported in Fig. 10(c) for the 60-kr/min case. In fact, lower rotor radius implies lower centrifugal forces, which reduces the iron rib thicknesses. It is worth to underline that the influence of the iron rib thickness is heavier on the q-axis inductance than on the torque, as shown in Fig. 10(d) and (e). The latter report the percentage variations of torque and q-axis inductance worsening always for the 60-kr/min case with respect to the design with no ribs. The high increase of the q-axis inductance causes a significant power factor drop, as reported in Fig. 10(f).
This analysis allows drawing the following considerations.
1) The presence of the structural iron ribs reduces the optimal split ratio and increases the optimal magnetic ratio since this design direction allows to minimize L q−rib [%] and so T rib [%].
2) The effect of the iron losses is that the optimum location moves toward the bottom-right of the design plane (i.e., higher split ratio and lower magnetic ratio) because, in the same direction, the magnetic anisotropy increases and the stator iron losses decrease. 3) When both iron losses and structural ribs are considered in the design exercise, the trends of the optimal split and magnetic ratios are a compromise between their competitive effects and so depend on the considered magnetic materials (as reported in Fig. 6).

C. Thermal Analysis
The comparative design exercise among different lamination materials has been carried out considering the same total losses equal to the maximum capability of the cooling system. This assumption guarantees that the thermal behaviors of the designed machines are approximately the same avoiding the need of redesigning a new cooling system for each motor. The total loss has been imposed considering the winding Joule loss at a given temperature (i.e., 130 • C) throughout the whole design process and independently of the lamination material. Although the total loss is kept constant, machines featuring the same constituent materials but different geometries (i.e., different sr and mr) or machines featuring different materials have a diverse loss distribution (copper, stator, and rotor iron losses). Consequently, the temperature distribution of the compared optimal machines will not be exactly the same even though the total loss has been kept unchanged. Indeed, machines with higher Joule losses experience higher winding temperatures as these are more difficult to be extracted with respect to the stator iron losses. Fig. 11(a) reports the winding temperatures of the optimal machines calculated with the commercial suite [44] considering an external spiral water jacket as a cooling system (whose details will be reported in Section VI). Fig. 11(b) shows the ratio between the Joule loss and the total stator losses. As expected, designs with higher Joule to iron loss ratio exhibit higher winding temperature.
In addition, machines designed with CoFe alloys (M4 and M5) have lower copper losses with respect to the SiFe ones (M2 and M3), and thus, they feature a lower winding temperature rise. Fig. 11(c) and (d) shows the rotor temperature and the ratio between the rotor iron loss and the total losses, respectively. Machines with mechanically optimized rotor material (M3 and M5) pay the price of higher rotor iron losses and therefore worse rotor temperature (compared to M2 and M4). Comparing solutions designed with CoFe and SiFe, the former definitely leads to higher rotor temperatures given their higher rotor iron losses. Fig. 11(e) outlines the power absorbed by the cooling system for all material combinations along with the one considered during the design process (black dotted line). It can be noticed that the dissipated losses deviate from the initial value due to the different winding temperature rise, leading to different Joule losses at thermal steady state. In any case, this deviation is within 10% confirming that the comparison is performed on a fair basis as the cooling system is suitable to cover all the considered machines. Fig. 11(f) reports the loss absorbed by the cooling system for the material combination M3 along with its stator and total losses. Up to a certain speed (approximately 50 kr/min), the losses absorbed by the coolant almost equal the total stator losses, while above this speed, the cooling system dissipates also part of the rotor iron losses. This confirms that the calculation of the maximum current with (4), considering also the rotor iron losses, is a conservative approach suitable for designing machines for a wide range of maximum speeds.

VI. EXPERIMENTAL VALIDATION
To validate the proposed design approach and the considerations regarding the soft magnetic material selection, a prototype of the optimal machine at 80 kr/min with the material combination M3 has been developed and tested. Before commencing the manufacturing, the rotor optimal geometry needs a structural FE refinement. In fact, the proposed design approach allows restricting the range of promising solutions, but once a geometry is chosen, its rotor needs to be structurally  optimized. Indeed, the distribution of the iron ribs along the barriers has to be optimized trying to keep the total iron bridge thickness per barrier as close as possible to the analytical estimated values. By doing so, the average torque would not change being not affected by the ribs position along the barrier but mainly by their thickness. A comprehensive description of the adopted structural design procedure is reported in [45], while Fig. 12 reports the Von Mises stress distribution of the final rotor geometry along with the photograph of the manufactured rotor lamination stack prior to the shaft and endcaps assembly. Fig. 13 shows a section view of the overall prototype highlighting the main components along with a photograph of the assembled system. The cooling system consists in a single spiral water jacket covering the whole stator axial length with a rated flow rate of 20 L/min [27]. Fig. 13 also shows the bearing housing featuring inlet and outlet channels for the air-oil mist lubrication of the selected bearings.
The prototyped SyR machine was coupled, via a gearbox (ratio 1:5.975) and a 3.5-Nm torque sensor, to a load motor (37-kW-20-kr/min IM) supplied with a four-quadrant regenerative drive. An in-house designed three-phase full-bridge converter featuring SiC power modules has been used to supply the machine under test. The control platform used to implement the control algorithm is based on a Xilinx Zynq7020 system on chip (SoC). A high ratio between the switching and the fundamental frequency was guaranteed setting the switching frequency to 40 kHz.
To confirm the FE-calculated performance and implement an accurate vector control, the flux-current relationships of the SyR prototype have been identified by adopting the procedure  presented in [46]. Fig. 14 reports the results of this experimental identification over the entire d-q current plane with a satisfactory agreement with the FE prediction.
After measuring the d-and q-axis fluxes, it is possible to derive the torque and inductances along the maximum torque per ampere trajectory needed to tune and implement the standard vector control scheme. After some preliminary tests, several motoring no-load tests have been performed by imposing a trapezoidal speed reference with increasing maximum values. Fig. 15 reports the measured and the reference speed, d-and q-axis currents, and reference voltages during a no-load test up to 35 kr/min. After verifying the safe operations up to a certain speed, an extensive campaign of generating load tests has been performed at different speeds and loads.
For every considered speed and load (up to 30 kr/min at full load, 28 A), the tests were carried out reaching the thermal steady state and acquiring several electrical, mechanical, and thermal variables. Fig. 16 shows the transient behavior of the speed, torque, voltage, and currents (in the d-q reference frame) during the current step between 21 and 28 A at 30 kr/min. Fig. 16 also reports the steady-state phase  currents, dc-link voltage, duty cycles, and torque. This condition corresponds to the machine producing the rated torque (T rated = 0.77 Nm).
Above 30 kr/min (at load) and 35 kr/min (at no load), the measured vibrations exceed the acceptable limits and, for the sake of safety, tests at higher speed have not been performed. An in-depth analysis of the vibration spectrum at the different rotating speeds leads to the conclusion that a shaft misalignment (or a nonperfect alignment of the bearings within their housings) is the most probable cause of this behavior. Fig. 17 outlines the measured temperatures experienced by the winding during the load tests (with torques from 1/4T rated to the rated value with a step of 1/4T rated ) at 30 kr/min. The thermal model used during the comparative design exercise described in Section V-C has been used to estimate the thermal performance of the prototyped machines at 30 kr/min and the rated torque. The average winding temperature predictions, also shown in Fig. 17, are underestimated (about 10%) with respect to the measured ones.

VII. CONCLUSION
This article has presented a hybrid design approach for SyR machines suitable for high-speed operation. Magnetic nonlinearity, rotor structural limitation, and the rise of iron losses have been carefully considered by weighing their competitive effects.
After FE validating the proposed approach, a comparative design exercise, investigating different soft magnetic materials, has been presented.
The analysis showed that adopting CoFe alloys leads to machines that outperform the SiFe counterpart up to a certain speed, which, for the considered case study, is about 60 kr/min. Above this speed, machines with SiFe provide better performance. In the lower speed range, the effect of the higher saturation flux density of the CoFe alloys allows achieving better performance. On the contrary, for higher design speeds, the higher iron losses and low yield strength of the CoFe materials outweigh the advantage of the higher saturation level and make the SiFe adoption more convenient. Using materials (either SiFe or CoFe) with a higher yield strength for the rotor always leads to better performance despite their higher iron losses. The analysis of the optimal machines showed that the split ratio and magnetic loads decrease as the speed increases with a higher decrement rate when adopting materials with low yield strength for the rotor.
The thermal analysis of the same set of machines has confirmed the validity of the implemented design approach. In particular, keeping invariant the total machine losses for all the design speeds allows a comparison on a fair basis, i.e., with the same cooling system. Although the distribution of the losses varies as the design speed increases, the maximum winding temperatures of all designs lie within an acceptable range, indeed.
Several experimental tests on a 6.5-kW-80-kr/min SyR machine have been reported by endorsing all the design considerations. Different mechanical limitations have constrained the maximum speed during the experimental tests up to 30 kr/min. Nonetheless, the reported experimental results up to the rated current fully validate the proposed design methodology.