High Speed Permanent Magnet Assisted Synchronous Reluctance Machines – Part I: A General Design Approach

The design of synchronous reluctance machines with and without permanent magnets assistance constitutes a challenging engineering task due to the numerous design variables and performance indexes to be considered. The design complexity increases even further when the application requires high speed operation, with consequent rotor structural constraints and related effects on the electromagnetic performance. Structured as two-parts companion papers, this work proposes a comprehensive design procedure able to consider all the non-linear aspects of the machine behaviour, greatly reducing the number of independent design variables, without worsening the computational burden. In particular, the non linear behaviour of the rotor iron ribs and the effect of the permanent magnets on the structural design are all taken into account with the proposed iterative design procedure targeting the achievement of a desired power factor. The proposed method will be then used to draw some preliminary design considerations highlighting the several trade-offs involved in the design of high speed permanent magnet assisted synchronous reluctance machine. Part I is setting the theoretical bricks that will be further expanded and experimentally validated in the companion paper Part II.

assisted synchronous reluctance machine, PMaSyRM) is far from being standardized as it involves multiple design choices with conflicting results [3]. The design task becomes even more challenging when the application requires high speed operations due to the increased complexity added by the rotor structural design [4], [5].
Two main strategies have been proposed in recent years to design PMaSyRMs. The first one divides the design process into two or more steps [6] while the other approach treats the electrical machine as a unique device [7].
With the first design approach, both stator and rotor geometries are identified following the design rules of SyRMs, thus disregarding the effect of the permanent magnet assistance [8]. After this first stage, the effects of the PM insertion are evaluated with analytical or finite element tools. In particular, the PM quantity is dictated by the adopted design criterion, i.e. obtain either the widest constant power speed range or the maximum torque increment [9], [10] subject to the constraints imposed by the demagnetization phenomena [11]. The distribution of the PMs within the rotor flux barriers is instead selected so to reduce the harmonic content of the resulting airgap flux density and consequently the torque ripple. Also this design step can be carried out analytically, as elegantly reported in [12], or by finite element method [13], which has the advantage of including all the secondary phenomena within the analysis (e.g. localized rotor saturation and slot harmonics effects). When designing PMaSyRMs for high speed applications, a further mechanical design step must be considered to determine the iron rib dimensions and distribution required to guarantee the rotor structural integrity at the maximum speed [14].
On the contrary, the second design philosophy treats the PMaSyRM as a unique device thus the PM electromagnetic and structural effects are considered from the initial stage. This approach has been implemented either with the aid of a simplified equivalent magnetic circuit [7] or by brute force FE-based design optimization [15]. In the first case, the hypothesis of linearity of the magnetic materials and the fact that the crosssaturation effect is neglected represent the main drawbacks. On the contrary, the FE-design optimization approach is very accurate but computationally more expensive than the analytical one. Albeit its high accuracy, FE-based optimization of multi-objective problems with a very wide research space often leads to sub-optimal solutions depending on the given time frame [16]. Indeed, unless a powerful computational cluster is available, dividing the research space and so the design procedure into several steps could lead to better results [17].
Although the comprehensive analytical design approach presented in [11], [18] is effective, at least as preliminary design stage to be followed by a FE refinement, it does not consider the permanent magnet influence on the stator design and neglects the soft magnetic material non-linearity. An interesting attempt to consider part of the non linearity effect has been proposed in [19] where the saturation of the main flux path (d-axis in Fig. 1) is considered with the aid of a non linear magnetic equivalent circuit. For high speed applications, when the dimensions of the rotor iron ribs become relevant, also the saturation of the high reluctance flux path (q-axis in Fig. 1) plays a major role in determining the machine performance. Indeed, the estimation of the q-axis inductance and of the PM flux linkage assuming a certain saturation level of the iron ribs looses accuracy as the speed increases, i.e. as the rib dimensions increase. Indeed, the PM insertion into the rotor flux barriers complicates both structural and electromagnetic aspects. In fact, the PM increases the rotor mass to be sustained by the iron ribs and the presence of the latter non-linearly affects the machine performance, i.e. PM flux linkage and q-axis inductance.
This work, structured as two-parts companion papers, extends the methodology presented in [20] in order to design PMaSyRMs suitable for high speed operations fully considering the effect of the PM presence and without neglecting the interaction between the structural targets of having iron ribs and their non linear electromagnetic behaviour. Indeed, the analytical performance prediction, used during the design, is enhanced by the calculation of the PM flux linkage and q-axis inductance with a non-linear lumped-parameters magnetic circuit capable of taking into account the saturation of the structural iron ribs leading to the correct sizing of the PMs which guarantee a specified performance requirement. The proposed design approach allows to define both stator and rotor geometries, which can be then FE-refined prior the manufacturing.
In the next section, the proposed analytical design procedure is described in depth. The following Sections III and IV report the FE validation of a wide range of designs along with a methodology to further improve the performance calculation accuracy. Section V provides some preliminary design considerations related to the optimal machine selection.
In the second part of this work, the proposed design approach will be used to identify the optimal geometries and related performance that can be achieved with and without the PM assistance as function of the speed. The outlined design considerations will be then experimentally validated showing the test results of a 8.5 kW-80 krpm PMaSyR prototype.

II. ANALYTICAL DESIGN
In this section torque and power factor are expressed as functions of two per-unit variables: the split ratio sr and the magnetic ratio mr, defined as: where r r is the rotor radius, r s the stator outer radius, B g the amplitude of the first harmonic of the airgap flux density and B fe the imposed iron flux density. The latter is the peak value of flux density within the stator yoke and it is chosen to be close to the knee point of the stator material BH curve, whereas the flux density in the stator tooth can be assumed to be equal to B fe or, as in this work, 20% higher. The remaining geometrical parameters, all depicted in Fig. 1, will be expressed in terms of these two independent quantities, allowing to fully define the machine geometry for a given outer envelope (i.e. outer stator radius r s and stack length L). In the next two subsections, first the main equations used to analytically estimate torque and power factor are recalled, then the non-linear q-axis equivalent magnetic circuit is described. The third subsection outlines different PM design criteria, while in the last part the iterative procedure required to calculate both PM flux linkage and q-axis inductance is described.

A. Main Design Equations
The torque produced by SyRMs or PMaSyRMs can be expressed, in its general form, as follows: while the internal power factor, defined by the sine of the angular displacement between the current and flux linkage vectors (Fig. 2), can be written as: where p is the number of pole pairs, i d and i q are d-axis and q-axis currents, whereas λ d and λ q are d-axis and q-axis flux linkages. For a given couple of independent design variables (sr, mr), the d-axis flux per pole is defined (2 · r r · L · B g /p), therefore the tooth width (w t ) and stator yoke radial thickness (l y ) can be evaluated imposing the iron flux density B fe [21]: where q is the number of slots per pole per phase. The stator tooth length (l t ) follows by simple geometrical considerations: where g is the airgap thickness. The complete rotor geometry, except for the PM dimensions, can be defined following the rules reported in [22]. On one hand, the barriers angular positions (in electrical degrees) at the airgap can be calculated so to obtain a uniform distribution of the equivalent rotor slots as in (7): where n r is the number of equivalent rotor slots whose choice depends on the number of stator slots if the minimum torque ripple condition is pursued. On the other hand, the i th barrier height h i a and surface S i b are uniquely defined imposing barriers having the same permeance (8) and the total iron thickness along the q-axis equal to the stator yoke thickness.
Dealing with PMaSyRMs, if the PM height is imposed equal to the flux barrier one, the only remaining degree of freedom is the PM width w P M . It is worth to underline that the above rotor design choices (which are just one option) allow minimizing the torque ripple [23]. Indeed, uniform distribution of the equivalent rotor slots and barriers with equal permeance allow obtaining a direct proportionality between the stator and rotor m.m.f. so to minimize their harmonic interaction, if the local saturation effects and the influence of different tangential ribs thicknesses are neglected.
The current components are calculated imposing the airgap flux density (B g and so the magnetic ratio mr) and the cooling system. In fact, the d-axis current can be inferred according to the Ampère's law: where N s is the number of turns in series per phase, k w is the winding factor, μ 0 is the vacuum permeability and k c is the Carter's coefficient. Conversely, the q-axis current can be computed knowing the maximum current I max which can be calculated imposing the cooling capability k cool as in (10): where k fill is the chosen slot fill factor, k cool is defined as the ratio between the allowed copper losses and the stator outer surface, ρ cu the copper resistivity, while L ew and A slot are the end-winding length the slot area which both follow from geometrical considerations. As just described, for a given split and magnetic ratio, it is possible to calculate the current components once the iron flux density B fe and the cooling system k cool are defined. In order to estimate both torque and power factor the flux linkages λ d and λ q need to be accurately calculated. The latter can be written as: where L dm and L qm are the d-and q-axis magnetizing inductances, L q−rib takes into account the flux short-circuited via the structural iron ribs, L dq accounts for the cross-coupling effects, L s is the leakage inductance and λ P M−d and λ P M−q are the dand q-axis fluxes produced by the PMs. Dealing with SyRMs, the last terms are clearly zero. Both L dq and λ P M−d terms are neglected in the analytical design because they can only be accurately estimated by means of FE analysis. On the contrary, many analytical formulations are available in literature to calculate all the components of the leakage inductance L s [24]. Regarding the magnetizing component of the d-axis inductance, it can be easily calculated neglecting the iron saturation as: where k w is the winding factor. These formulations ( (9) and (13)) lead to an overestimation of the d-axis inductance while the d-axis current is underestimated. This limit can be overcome by introducing a factor (k d−sat > 1) which takes into account the saturation of the d-axis magnetic flux path. The evaluation of k d−sat is performed by solving a simplified equivalent magnetic circuit as proposed in [21]. A q-axis equivalent circuit accounting for the non linear behaviour of the iron ribs is presented in the next subsection. It is worth to underline that the stator and rotor geometrical parameters are calculated disregarding the permanent magnets flux. This is clearly valid for SyRM; however, when designing PMaSyRMs, this assumption remains valid as long as the PM flux linkage does not substantially affect the total flux. This detail will be discussed thoroughly in Section V.

B. Non-Linear Q-Axis Equivalent Magnetic Circuit
If only the q-axis component of the stator m.m.f. is considered, the rotor iron segments assume different magnetic potentials. In other words, the rotor magnetically reacts to the applied stator m.m.f.. The no-load flux linkage due to the PMs and the q-axis inductance can be calculated solving the equivalent magnetic circuit reported in Fig. 3(a) for a rotor with 3 barriers per pole. In the latter, the m.m.f generators F 1 , F 2 and F 3 stand for the qaxis stator magneto-motive forces averaged along the periphery of the rotor with intervals equal to the rotor pitches Δα i as shown in Fig. 4. The reluctances related to the airgap paths can be calculated for the i th equivalent circuit branch as: The PMs are represented by the series of the m.m.f. generators U i m and the reluctances R i m which can be both calculated for the i th barrier as: where B rem is the residual flux density of PM material which is set to zero when designing SyRM and μ P M is the relative permeability of the PM. The remaining part of the barrier is taken into account by the reluctances R i a equal to: where the radial height of the ribs h i r is here considered equal to the respective barrier one h i a while their thickness w i r follows from mechanical considerations.
Indeed, the iron rib thickness of the i th barrier can be computed adopting a simplified analytical model which considers only the centrifugal force F i c acting on the relative flux guide and PM [25] as shown by the shaded region in Fig. 4(b): where k s is a safety factor (usually between 2 and 3), σ s is the yield strength of the rotor lamination, m i fg is the flux guide mass and m i P M the PM masses which have to be sustained by the i th iron rib, R i cg is the coordinate of the center of mass of the considered region, and ω m is the mechanical speed.
The iron rib permeability μ i rib−dif f and the flux density B i rib0 both depend on the rib working point B i rib on the non-linear B-H curve of the rotor soft magnetic material as depicted in Fig. 5. Indeed, they are defined by the line tangent to the point B i rib which can be expressed as: (21) from which it is possible to calculate both φ i r and R i r required to solve the equivalent magnetic circuit.
It is a common practice to consider all the iron ribs equally saturated at a predefined value B rib whatever operating condition is analyzed. This approximation is acceptable as long as the dimensions of the ribs are relatively small and the shunted flux does not substantially modify the machine performance. Clearly, this assumption looses its validity as the speed increases since the iron ribs dimensions become more important. In fact, as the speed increases the effective working point B i rib considerably deviates from the a-priori defined value; moreover each iron rib may assume different operating points. Consequently, an iterative procedure is required as the speed increases in order to update the parameters defining the iron ribs behaviour, i.e. the flux generators φ i r and the reluctances R i r . Furthermore, when the PMaSyRM case is considered, these values are also affected by the amount of PM material placed within the rotor slots and their distribution.

C. PM Design Criteria
The q-axis equivalent magnetic circuit reported in Fig. 3(a) can be simplified applying the Thèvenin theorem; Fig. 3(b) shows the equivalent of one branch of the circuit, where R i eq is the parallel between R i a , R i m and R i r , while U i eq can be computed as: Writing the first Kirchhoff's law at the nodes 1, 2 and 3 in Fig. 3 and expressing the flux flowing in the connected branches as a function of the m.m.f. related to the PM and ribs (U eq ), stator excitation (ΔF) and rotor reaction (ΔZ), it is possible to write the following matrix equation: where A, B and C are matrices defined by the barriers, airgap, magnets and ribs reluctances while the three vectors U eq , ΔZ and ΔF are defined as follows: As shown in (22), U i eq depends on the PM widths which have to be identified according to a specific criterion. One possible option is to design the PM dimensions in order to maximize the flux weakening capability of the machine and so its constant power speed range. It is well known that this condition is achieved when the PM flux linkage equals the q-axis flux at the rated condition [26]: where λ P M−NC is the desired PM flux linkage and L q is the total q-axis inductance (sum of the magnetizing and leakage components). An elegant way to implement this PM design criterion, also known as natural compensation ( NC ), is to impose the fluxes entering the branches of the stator m.m.f generators (reported in Fig. 3(a) with the symbol F i ) equal to zero. In other words, the rotor magnetic potentials (Z i ) have to equal the stator ones (F i ). Imposing the condition ΔZ = ΔF allows to determine the equivalent m.m.f. U eq : from which the PM widths w i P M−NC can be calculated: supposing a certain iron rib dimension w i r and electromagnetic exploitation B i rib . It is worth to underline that adopting this PM sizing approach along with the criterion of designing the rotor flux barriers with equal permeance, allows minimizing also the torque ripple. In fact, when these two criteria are adopted, the matrix B −1 · (A − C) is diagonal, which means that the equivalent PM m.m.f. (U eq (α) in Fig. 4) is proportional to the stator m.m.f F (α). Therefore the harmonic interaction between these two m.m.f. is minimized.
For this reason, if another PM design method is adopted, it is convenient to simply scale the PM dimension obtained with the natural compensation criterion (w P M−NC ) in order to keep the proportionality between stator and PM m.m.f. For example, if a certain value of internal power factor is desired (ipf * ), the required PM flux linkage λ * P M−q , neglecting both cross coupling terms (L dq and λ P M−d ), would be: Once the PM flux linkages are known in the natural compensation condition λ P M−NC and in the condition providing a given internal power factor λ * P M−q , the new PM dimensions can be simply calculated as: Once the PMs widths are defined, the q-axis equivalent magnetic circuit, reported in Fig. 3, can be solved to compute the no-load PM flux linkage and the q-axis inductance.

D. Iterative Calculation of the Q-Axis Inductance and PM Flux
Based on the considerations reported in the previous subsections, the design of the PM dimension and the calculation of the q-axis quantities require an iterative procedure. In fact, the iron rib dimensions (w i r ) structurally depend on the PM width w i P M (via the mass m i P M in (20)), which in turn is electromagnetically affected by the iron ribs width as shown in (27). The last equation also proves that the PM dimension is affected by the electromagnetic working point of the respective rib, i.e. from the two parameters B i rib0 and μ i rib−dif f included in R i eq . The latter follows from the resolution of the equivalent circuit which clearly depends also on the iron rib widths.
A comprehensive iterative procedure, summarized in the flowchart reported in Fig. 6, is here proposed to solve the non-linear q-axis magnetic circuit considering all the above mentioned dependencies between structural and electromagnetic design aspects. The SyRM design is a particular and simpler case of the PMaSyRM one. In particular, the workflow is constituted by the following steps.
1) For each machine defined by the two independent variables sr, mr, the design of the main stator and rotor dimensions is carried out (including the barrier height h i a and barrier surface S i b , tooth width w t and tooth lenght l t , and yoke length l t ) without considering the PM assistance as described in Section II-A. 2) Then, the iron ribs dimension are calculated using (20) without considering the PM weights and supposing a certain electromagnetic exploitation of the ribs, i.e. B i rib so to be able to determine both B i rib0 and μ i rib−dif f needed to define their circuital representation. 3) Consequently, if a PMaSyRM is considered, the PM dimensions can be calculated (25)- (27) once the PM design criterion is selected. If the max ipf criterion is chosen, the q-axis inductance L q required to evaluate λ * P M−q and so the PM widths w * P M is estimated at the first iteration using the approximate formulation reported in [21], [22] which does not require the resolution of the equivalent circuit. Otherwise (i.e SyRM case), the algorithm proceeds directly to step 5. 4) Once the PM widths are known, the iron ribs dimensions are re-calculated taking into account the PMs masses and compared with the previous values. If their difference lies within predefined bounds, the algorithm proceeds to the next step; otherwise, the PM widths are re-calculated considering the updated values of iron ribs w i r . 5) Once both PM and ribs dimensions are defined, the magnetic circuit is fully defined and it can be solved calculating the unknown rotor reaction ΔZ via the (23). The effective saturation level B i rib−new in Fig. 5 can be evaluated for each iron rib with the following relationship: The so computed value is then compared with the previous one (B i rib ). If their difference is acceptable the algorithm proceeds to the next step; otherwise, the new value B i rib−new is used to update the parameters of the tangent line (B i rib0 and μ i rib−dif f ) depicted in Fig. 5 and the algorithm restarts from the step 2 in the PMaSyRM case (re-calculating the PM dimensions) or step 5 in the SyRM case. 6) Once the saturation level of each rib is correctly evaluated, it is possible to fully define the variables related to the q-axis, i.e. the PM flux linkage and the q-axis inductance, considering either the PM or the stator excitation. If this calculation step is performed still modeling the iron ribs as the parallel between the flux generator φ i r and the reluctance R i r , then the rib working point (B i rib ) would move on the line defined by the parameters (B i rib0−new and μ i rib−dif f −new ) depicted in Fig. 5. However, this approach would lead to incorrect estimation of λ P M and L q since the electromagnetic rib working points would differ from the load scenario (when both stator and PM excitations are present). To avoid this issue, the PM flux linkage and the q-axis inductance are calculated modeling the ribs with a single reluctance R i r−app defined by the apparent permeability (μ i rib−app ) as shown in Fig. 5: By doing so, the electromagnetic working points of all iron ribs are unequivocally fixed to the one related to the load scenario. Therefore, the calculation of the q-axis inductance can be performed by modeling the iron ribs only with the apparent reluctances, turning off the PMs m.m.f. U i m and solving the resulting Thévenin magnetic circuit. Indeed, the resolution of such circuit leads to the calculation of the rotor magnetic potentials ΔZ i . From these values it is possible to calculate the airgap flux φ i g flowing in each airgap branch of the magnetic circuit and, therefore, the airgap flux density B i g . Then, the complete waveform of B g can be evaluated by combining all the calculated B i g (three in this case). From the obtained waveform, the first harmonic B g1 is calculated and used to compute the flux per pole due to the q-axis current contribution and the inductance L q as in (32): where τ p is the pole pitch in meters.
The same workflow can be adopted to calculate the flux per pole due to the PMs, by turning off the stator m.m.f. generators and applying the above procedure for the airgap flux density computation. 7) As last check and only for the PM-assisted machines, the computed L q is compared with the one used to estimate the PM flux linkage needed to achieve the desired internal power factor (28). If these two values are similar the algorithm ends, otherwise it restarts from step 2.

III. FE VALIDATION OF THE ANALYTICAL MODEL
The proposed analytical design approach has been implemented considering a wide range of design variables sr − mr. Table I summarizes the main assumptions and constraints. As previously pointed out, the outer envelope is fixed (i.e r s and L are given).
The airgap thickness has been selected as a compromise choice between the need of mimizing it (beneficial from the electromagnetic point of view) and practical assembly limits, whereas the selected number of pole pairs allows a lower fundamental frequency beneficial for both control and thermal management. The stator and rotor materials are chosen so to minimize the iron losses and improve the rotor integrity at high-speed (further details about the materials selection can be found in [27]); the iron flux density is a consequence of the selected iron material for the stator lamination.
The PM dimensions have been designed to achieve an internal power factor ipf * equal to 0.9. The estimated performance of each machine of the design plane sr − mr has been compared with the respective FE analysis with the aim of investigating the accuracy of the analytical approach.   Fig. 7 reports a comparison between the analytical ( an ) and FE computation ( F EA ) of d-axis inductance ( Fig. 7(a)), q-axis inductance ( Fig. 7(b)) and PM flux linkage (Fig. 7(c)). The empty areas of the design plane are due to the presence of unfeasible machines. The FE calculation of these three parameters have been performed with the frozen permeability method [28] in order to capture their values in the load scenario (i.e. when supplying with i d , i q and PM excitation). Analysing Fig. 7 it is worth to underline the good estimation of all parameters from a qualitative point of view. The excellent agreement of both q-axis inductance and PM flux linkage can be ascribed to the good estimation of the iron ribs saturation levels. Fig. 8 reports a comparison between the analytical and FE calculations of B rib of each flux barrier. It is worth to notice that the contour plot of the outermost rib features an additional empty zone (left area of the design plane): the machine of such zone are still feasible but the respective outermost ribs width is zero or falls within the mechanical tolerance.
Although inductances and PM flux linkage are estimated with a good degree of accuracy, the torque, highlighted in Fig. 9(a), shows non-negligible discrepancies respect to the FE-computed values. These residual errors are mainly due to the non-linear   phenomena not considered in the analytical model, namely the cross-saturation effects (i.e L dq and λ P M−d ). Fig. 10(a) reports the percentage rate between the d-and q-axis PM flux linkage, whereas the influence of the cross-coupling contribution is shown in Fig. 10(b) in terms of percentage rate between L dq and L dm . The torque contour differs from the FE one mainly in the zone of the sr − mr plane characterized by large cross-saturation phenomena. As a consequence, the inclusion of such effects in the design stage is needed to reach an accurate prediction of the electromagnetic performance of both SyRM and PMaSyRM. Conversely, the analytical ipf contour shows a good agreement with the FE one, both in terms of numerical values and contour shape. It is worth to underline that the obtained internal power factor is not the desired one (i.e. 0.9) over the entire sr − mr plane. In Section V the reason behind this behaviour will be fully investigated.

IV. ENHANCING THE DESIGN APPROACH ACCURACY
With the aim of improving the performance estimation without sacrificing the fast evaluation characteristic of the pure analytical approach, a hybrid design procedure, proposed in [20] for SyRMs, is extended here to the PM-assisted ones.
In particular, once the analytical design is performed for the whole sr − mr plane, the 4 machines at the corners of the plane are selected and FE-simulated. From the FE-simulation all inductance (L dm , L qm , L dq and L s ) and PM flux (λ P M−d and λ P M−q ) contributions can be correctly evaluated adopting the frozen permeability method [28], [29]. This allows the calculation of several correction factors defined as the ratio between the finite element ( F EA ) computed value and the respective analytical ( an ) prediction as: where x can be either the d-or q-axis PM flux linkage λ P M−d , λ P M−q , 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. The so computed correction factors are then extended to the entire design plane sr − mr using a linear interpolation, allowing the adjustment of the magnetic model. By doing so, the cross-coupling terms (L dq and λ P M−d ) are taken into account within the design workflow and the mismatches between analytical and FE-computed magnetizing inductances and q-axis PM flux are greatly reduced as shown in Fig. 11. The good agreement in terms of d-and q-axis inductances and PM flux linkage leads to the excellent matches between the adjusted torque and internal power factor contours shown in Fig. 12.

V. PRELIMINARY DESIGN CONSIDERATIONS
The reduction of the design space to only two variables (sr, mr) and the accurate calculation of torque and power factor as their function greatly reduce the design complexity. Indeed, the selection of the machine to be considered for the final design refinements (needed prior the prototyping) is greatly simplified.  The choice of the final design is not univocal and clearly depends on the application requirements, e.g. the performance indexes having the priority (e.g. maximum torque, maximum power factor, minimum PM volume, etc.). In the following, the contours of the variables influencing mostly torque and power factor are analysed in order to infer the different trade-offs involved when designing high speed PMaSyR machines. Fig. 13(a), (b) and (c) report the constant loci in the sr − mr plane of total torque, PM torque and reluctance torque component, respectively. The total torque contours and so the location of the maximum torque design (showed with the marker ) depend on the concomitant actions of the reluctance and PM torques. Fig. 13(d), (e) and (f) show the contours of the variables defining the reluctance torque behaviour, namely the d-and  q-axis currents and the inductances difference, respectively. It can be seen that the reluctance torque, and the location of the maximum reluctance torque design in the sr − mr plane, is the compromise between the need of maximizing the magnetizing current (i d ), the anisotropy (L d − L q ) and the quadrature current (i q ) [20], [22].
The shape of the PM torque component contours depends on both d-axis current (Fig. 13(d)) and PM flux linkage ( Fig. 13(g)). The latter is heavily affected by the amount of PM material imposed by the selected PM design criterion, as it can be seen comparing Fig. 13(g) and (h). The PM volume V P M is also affected by the maximum available space within the rotor flux barriers (V slots ), shown in Fig. 13(i). Fig. 14(a) and (b) report the internal power factor of the PMaSyR machine and the respective value obtained without the PM assistance, respectively. Although the PM-assistance greatly improves the internal power factor respect to the pure reluctance machines, only part of the design plane sr − mr provides the target internal power factor (i.e. 0.9). For some region of the design plane, the PM volume required (V * P M ) to satisfy the selected design criterion exceeds the available space within the rotor flux barriers. Fig. 14(c) reports the ratio between the effectively used PM volume and the ideally required value justifying this unexpected behaviour. In particular, the ipf values differ from the target one when designing machines with high magnetic ratio and low split ratio, as shown in Fig. 14 a. Indeed, higher mr implies a higher thickness of the flux guides which reduces the flux barriers radial thickness (h i ) and so the available space for the PMs. Similarly, lower sr implies a reduction of the rotor diameter and so also of the flux barriers area. Lastly, Fig. 15 shows the percentage difference between the module of the total flux and its d-axis components with respect to the latter. This figure of merit quantifies the approximation related to the assumption of neglecting the PM flux during the design of the stator tooth, yoke and rotor flux guides thicknesses. Analysing this figure, it is straightforward to conclude that the PM effect can be surely neglected during the magnetic design of the stator and rotor laminations. However, this does not imply that the PM presence should not influence the selection of the final machine. For example, selecting the design providing the maximum reluctance torque (shown with in Fig. 13(c)) and adding the PM within the rotor slots (according to a certain criterion) would lead to a sub-optimal design. In fact, the resulting machine would produce a lower torque (1.1 Nm in this case) compared to the maximum torque design calculated considering both PM and reluctance components (1.4 Nm showed with in Fig. 13(a)).

VI. CONCLUSION
This work has presented a general and comprehensive design procedure for high speed permanent magnet assisted synchronous reluctance machines. The proposed analytical design workflow allows considering all the non-linear aspects of the machine behaviour reducing to two the number of independent variables.
In particular, the resolution of the q-axis equivalent magnetic circuit has been proposed to calculate both q-axis inductance and PM flux linkage taking into account the non-linear behaviour of the rotor iron ribs. Then, an iterative procedure has been proposed to design the PM dimensions, to achieve a desired power factor, and to calculate the q-axis performance taking into account the electromagnetic non-linear behaviour of the iron ribs which are sized also considering the mechanical effect of the additional weight of the PMs. After FE validating the analytically predicted torque and power factor, a computational efficient FE-based approach has been implemented in order to enhance the accuracy of the analytical performance estimation.
The proposed hybrid analytical-FE design approach has been then applied to a specific case study to draw some preliminary design considerations. It has been shown how the PM-assistance can be neglected during the magnetic design of the stator and rotor laminations, and how this heavily affects the selection of the optimal machine.
In the next part of these two companion papers, the effects of the PM insertion when designing PMaSyRM with increasing maximum speed and different PM types will be analyzed in depth. Particular attention will be paid to the optimal machine selection and how this choice affects the maximum power capability and the overall geometry with respect to the operating speed. Then, all the FE refinements stages carried out before the manufacturing of the selected design will be shown along with the experimental test results.