Homothetic Design in Synchronous Reluctance Machines and Effects on Torque Ripple

This paper presents a novel design concept for Synchronous Reluctance (SynRel) machines aimed at reducing the torque ripple. Two general sizing approaches based on the homothetic scaling principle are defined and compared. An in depth analysis on the torque ripple, for a wide range of scaled geometries, evaluated by finite element, has been carried out at different operating conditions. A further analysis is performed on 4 scaled geometries that have been optimized starting from 4 random rotor geometries. It is shown that the main rotor geometrical variables converge to similar values for all scaled machines. The accuracy of the proposed model is then validated by comparing the FE simulated torque ripple waveforms with the experimental data carried out, for a range of operating conditions, on a machine prototype. The outcome of this work is a fast and accurate scaling technique for the preliminary design of SynRel machines with reduced torque ripple.

these advantages, the main pitfall of this machine topology is the conspicuous torque oscillation, which is an undesired torque component causing acoustic noise, vibration and may degrade the drive controllability. Several techniques for the torque ripple reduction have been investigated over the last two decades and they can be classified into two major categories. The first one acts on the control scheme [1]- [3], while the second consists of specifically tailored motor-design techniques [4], [5]. The first approach is more broadly applicable, but it complicates the control algorithm structure and so its computational cost. While, the second approach obviously requires the development of new machine designs and this is not always possible. Several design techniques have been proved effective in minimizing torque oscillations, such as suitable choice of the flux barriers with respect to the number of stator slots [6], suitable flux barrier angular displacement [7], [8], rotor skewing [9], etc. The proposed design guidelines originate from considerations based on analytical models, which often rely on a set of hypotheses introduced to simplify the analysis, and to make it feasible. Such analytical models most of the time neglect the effect of the non-linearities and geometrical complexities on the predicted performance. Therefore, they are useful only during the preliminary design stage. The next refining stage is then carried out by means of finite element (FE) analysis, which is able to evaluate the design aspects disregarded in the first stage (e.g., non-linearities heavily affect the torque profile). During the detailed design phase, several iterations are required and the computational cost depends also on the accuracy of the analytical model used in the preliminary design. Clearly, the more the analytical model is able to predict the machine performance faithfully, the less FE iterations are needed in the second design stage. Indeed, a more accurate analytical model is able to better identify the design space area to further explore via FE analysis. The second design stage is commonly implemented as a FE-based design optimization. Several works have addressed the problem of further reducing the computational burden required to carry out the optimization, which depends on two factors: the computational time required to evaluate the performance of a single machine candidate and the geometrical complexity of the machine structure to be optimized. The computation time varies according to which performance indexes are being optimized (torque, torque ripple, iron losses, etc.) and [4], [5], [10], [11] have been investigated the problem reaching a good trade-off between accuracy and computational burden. On the other hand, the geometrical 0885-8969 © 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
complexity of the problem can be further reduced acting on how machine geometry under investigation is parametrized. In particular, [11] and [12] present a comparative study among different SynRel rotor flux barrier parametrizations, analysing the compromise between geometrical complexity and achieved performance. It is a general conclusion that adopting a flux barrier profile described by the Joukowski equation and a flux barrier parametrization described by three parameters (barrier thickness, air gap angle and end-barrier parameter) is the best compromise between performance and geometrical complexity [12], [13]. These parameters are also the ones which most affect the torque performance, and for this reason they usually optimized during the FE refinement. The purpose of this paper is to show how the FE design stage can be greatly simplified and so computationally relieved by considering a novel dimensioning homothetic approach during the first analytical design step. The homothetic scaling design principle was initially introduced for the induction motors in [14], and for permanent magnet synchronous motors in [15]. In this paper this concept is proposed for SynRel machines based on the analytical model presented in [16]. This work, which is a continuation of the authors' previous research on homothetic scaling for the design of synchronous reluctance machines [17], addresses the effect of different scaling approaches on the torque ripple.
The paper is structured as follows. In Section II the reference machine geometry is defined along with the homothetic design scaling principle. Then two scaling methods are assessed in Section III and different geometries are evaluated over a wide range of operating conditions, by means of FE simulations. In Section IV, four scaled machines are optimized minimizing their torque ripple and to show the correlation between the initial scaled geometry and the optimized one. The design approach is then validated in Section V, comparing the torque ripple of the scaled machines, computed by means of FE simulations, against the experimental measurements on the reference machine prototype for different operating conditions.

II. SCALING PRINCIPLE AND REFERENCE MACHINE DESIGN
In the following two subsections, the preliminary sizing method of the reference machine is outlined along with the scaling principle. The sizing approach has been extensively described in [16], where the anisotropy of the rotor is considered as an input of the design procedure.

A. Preliminary Sizing of the Reference Machine
The general sizing approach for a SynRel machine starts from the well-known torque equation written in the classical synchronous (d-q) reference frame: where L d and L q are direct and quadrature inductances and i d , i q are direct and quadrature currents, whereas p is the number of poles. K dm , K qm are d-q axes magnetizing coefficients, which are related to the salient nature of the reluctance machines and can be derived as: As presented in equations (2)-(3), B 1 represents the fundamental component of the air-gap flux density for a uniform air-gap machine (no saliency) and B 1d , B 1q are set to be the fundamental flux-density components along the d and q axes under produced by the same stator ampere-turns. Hence, ratios of fundamental flux density components are defined as magnetizing coefficients. Using equations (1)-(3) the saliency ratio can be derived as proposed in [16]: where L l represents the leakage inductance. Based on equations (2)-(4) the saliency ratio ξ and magnetizing coefficients can be analysed using permeance functions along d and q axes described in [16]. For a 3-phase distributed winding cylindrical machine, the magnetizing inductance L m is calculated as shown in (5), where R ro is the rotor diameter, L stk is the stack length, q is the number of slots per pole per phase, g is the air gap length, μ 0 is the relative permeability of air. K w1 and K s are winding factor for the fundamental component and saturation coefficient, respectively, and n s -is the number of turns per phase [18]: In (5) the parameters R ro and L stk are the variables of interest as these determine the size of the machine. In order to relate R ro and L stk, the aspect ratio γ can be used (6): The torque equation (1) can be further expanded using the equation (2)-(6) as it was shown in [16]: Fig. 1 reports the flow chart of the adopted sizing approach, which includes 5 steps. Starting from the performance requirements and the design constraints, the second step defines the initial guess values of the machine's saliency ratio ξ, the rotor's magnetic insulation ratio k air [16] and the number of barriers k. Using all predefined parameters above, the machine is sized using equation (7) during the third step. The saliency ratio is then estimated with an analytical model based on the equations (2)-(4) [19] and then the electromagnetic torque is calculated with equation (1). The machines' torque can be tuned in an iterative fashion by either varying k, k air or the main rotor diameter D ro , depending on the performance specifications.  The SynRel machine used in the following as reference design has been initially sized for a household appliance application whose design specification and constraints are listed in Table I. The winding layout was designed based on voltage-speed limit and the current density requirements [20]. Table I also reports the main dimensions and the winding details of the reference machine.

B. Rotor Design of the Reference Machine
The rotor geometry of the reference machine has been optimized to be suitable for both reluctance and permanent magnet assistant reluctance variants. For this reason, the rotor barriers are presenting a central rectangular slot, to host permanent magnets if needed, as shown in Fig. 2. The optimisation has been carried out considering the rotor parameters shown in Fig. 2 (i.e., barrier angles and thicknesses) and the optimization procedure is fully described in [21].

C. Scaling Principle
Based on the homothetic scaling principle discussed for Induction Machines in [14] as well as for the SynRel in [17] the reference machine can be scaled both radially and axially.
In the following, the aspect ratio of the scaled machines will be kept constant, therefore the stack length L stk will be scaled proportionally to outer rotor diameter D ro . The radial scaling can be carried out pursuing two approaches, i.e., keeping fixed the airgap length (AGF) and scaling the airgap length (AGS) with the same factor of the cross-sectional parameters. Equation (8) and (9) describe the scaling factors for the fixed airgap approach: where D si-n is the stator inner diameter of the scaled machine, while D si-ref is the stator inner diameter of the reference machine. Clearly, such approach utilizes two different scaling factors, S si for the stator and S ro for the rotor, while when scaling the airgap length as well, the scaling coefficients is the same for both stator and rotor (10):

III. EVALUATION OF THE SCALED MACHINES TORQUE PERFORMANCE
In the following two subsections, the torque ripple of several scaled machines is FE evaluated for different operating points in the d-q axis current plane. In particular, in the first subsection two scaled machines are considered, one obtained keeping fixed the airgap length (M3) while the second (M2) also scaling the latter. Table II summarises the geometrical parameters featured by the scaled machines (M2 and M3) and the reference one (M1). In subsection III-B, the same analysis is extended to a wider range of scaling factors for both AGS and AGF cases, respectively.   It can be observed that the iso-curve of average torque does not vary significantly, which implies that the maximum torque per ampere (MTPA) locus is almost the same for the three machines as shown in Fig. 4 (y-axis represents the current excitation angle γ o = tan −1 (I q /I d )). The latter reports the current phase angle corresponding to the MTPA condition as function of the current module for the reference and scaled machines. The average  torque produced by the M2 geometry is lower compared to one obtained with the M3, i.e., at rated current density of 5 A/mm 2 the average torque achieved by M2 T M2 = 5.6 Nm whereas T M3 = 6.1 Nm; this is clearly due to the bigger airgap of the M2 geometries respect to the M3 one.

A. FE Evaluation of M1, M2 and M3 Geometries
The torque ripple contours of the scaled geometries M2 and M3 follow the same pattern featured by the reference geometry (M1) as shown in Fig. 3(b), (d) and (f). Fig. 5 reports the percentage torque ripple of the three considered machines at the MTPA condition as function of the current amplitude. The torque ripple of the scaled M2 geometry follows almost the same pattern of the reference geometry M1 except for the really low and high current modules. This is due to bigger air gap; hence it requires higher current to properly saturate the ribs, as the machine was geometrically scaled. The scaled machine M3 shows a higher torque ripple respect to the reference geometry M1 due to different scaling of rotor and stator. Based on all the above it can be concluded that both scaled machine, M2 and M3, feature a torque ripple comparable with the base geometry. In particular, the torque ripple variations lie within a 15% range over a wide range of currents. As can be observed at rated current density of 5 A/mm 2 , the M2 has T ΔM2 ∼ 10.9% whereas M1 shows T ΔM1 ∼ 11%; M3 shows relatively higher torque ripple T ΔM2 ∼ 15%.

B. FE Evaluation of Wide Range of Scaled Geometries
A total of 9 machines have been obtained within the range of 0.5 ≤ S si ≤ 4 by scaling the reference geometry M1 adopting both AGS and AGF approach. Fig. 6(a) and (b) report the percentage torque ripple at MTPA condition in terms of contour in the plane stator inner radius -phase current.
Analysing the torque ripple of the machines uniformly scaled (AGS), it can be noticed that in for low current values (i.e., 5A ≤ I s ≤ 20A), the torque oscillation remains within the range 10% ≤ T Δ ≤ 17% for all the considered radial dimensions and current loading. On the contrary, the torque ripple shown in Fig. 6(b), related with the AGF geometries, show a significant increment compared to the reference machine M1.
A torque ripple within the range 10% ≤ T Δ ≤ 17% is obtained only for machine having 0.5 ≤ S si ≤ 2. It can be concluded that the AGS scaling approach leads to a moderate torque ripple variation over a wider range of scaling factor, whereas adopting the AGF scaling approach, the torque ripple variation is more pronounced.

IV. TORQUE RIPPLE OPTIMIZATION
The following exercise aims at demonstrating that starting from a random set of rotor parameters, the optimization algorithm converges to an optimal rotor with a geometry similar to the reference one. In order to demonstrate the above statement and the differences between AGS and AGF scaling approach, 4 different scaled machines have been considered and optimized.
The first two (M2 * and M3 * ) are the machines considered in Section III-A and whose parameters are detailed in Table II. The other two scaled machines (labelled as M4 * and M5 * ) are obtained from the reference one adopting a scaling coefficient lower than one (S si = 0.75). The geometries M2 * and M4 * are obtained via the AGS approach while the M3 * and M5 * using the AGF method.
The geometrical variables to be optimized are the angles defining the barrier position at the airgap. Table III reports the lower and upper limits that those variables can assume during the optimization while the stator geometry remain fixed. The insulation ratio, defined as the per unit air portion of flux barriers along the q-axis: where D sh is the shaft diameter and hc k is the kth barrier thickness (as shown in Fig. 2), which is kept constant during the rotor optimization process.
The choice of keeping the insulation ratio (k air ) invariant during the optimization is related with the need of obtaining machine producing approximately the same average torque. Indeed, it has been demonstrated that the insulation ratio has a bigger impact on the average than the flux barrier angles as it has been shown in [22].
With the choices motivated above, the optimisation problem presents a single objective function, the torque ripple. An heuristic optimiser (simplex algorithm) has been adopted to carry out the FE-based design optimization whose workflow is shown in Fig. 7. The initial Design of Experiments table used to start the search has been defined by a random sequence. The number of individuals for each generation has been set to 60 and a maximum of 10 generations has been considered leading to a total of 600 functional evaluations. An automatic drawing  and solving procedure has been implemented via Matlab and the finite element software FEMM 4.2. The torque ripple (at MTPA condition with a current density of 5 A/mm 2 ) of each machine candidate is determined by a series of static simulation performed uniformly over one torque ripple period. Fig. 8 shows how the geometrical variables converge to the optimal values leading to the minimum torque ripple for the scaled machines M2 * , M3 * , M4 * , M5 * . It can be clearly observed that the trends of the barrier angles converge approximately to the same angles. The summary of optimal angles is reported in Table IV. Based on the convergence of the angles value, it can be noticed that the variations of the θ 2 and θ 3 are not significant, all within a range of 0.4 o , whereas the difference in θ 1 is significant only for M3 * geometry. This can be explained It can be concluded that the homothetic scaling, starting from a well-designed and optimized reference geometry, lead to a scaled design which is a solution that can be considered optimal, or for sure a good starting point for further torque ripple optimization refinements. Consequently, the design variable boundaries can be greatly restricted relieving the computational burden of the FE refinement design stage.

V. MECHANICAL CONSIDERATIONS
In this section other mechanical aspects not previously considered are discussed. The thermal behaviour of the electrical machine is mainly a function of the current density, as well as the cooling type that is adopted by the system [21]. The current density was kept constant for all 5 machines including M1, therefore current was proportionally scaled, as the area of the slot is increased or decreased. As shown in Table V, the area of the slot is scaled by S si 2 . Electric loading A s is highlighted in Table V to illustrate the difference among the analysed motor variants.
In a SynRel motor, the design of both radial and tangential ribs has been investigated extensively [21], [23], [24]. The function of the iron ribs is to mechanically retain the rotor parts together and to withstand the centrifugal force depending on the speed of the machine. Hence the ribs thickness is mainly affected by the maximum speed and the rotor geometry. For example, if the scaling leads to thinner ribs the maximum allowable speed of the machine could be affected and a mechanical refinement is required to guarantee the structural integrity of the rotor. In the presented homothetic method, the ribs have been scaled proportionally. This is valid within certain scaling range. Fig. 9 presents the FE simulated mechanical stress maps for 4 scaled geometries S si = 0.75, 1.5 at n = 18000 rpm. Mechanical FE simulations were carried out considering the parameters of the original geometry M1 as shown in Table VI, with highlighted ribs thicknesses according to Fig. 2 and physical properties of the silicon steel used. As can be observed the smaller scaled geometries S si = 0.75 have the peak stress at the ribs which is within the allowable value of the σ stress , whereas the S si = 1.5 scaled geometries are close to the yield stress σ stress .   Fig. 10 presents the results for a wider speed range based on FE simulated mechanical stress test of 7 different geometries: 3 geometries were scaled based on (8) and (9) (AGF) and 3 geometries were scaled using (10) (AGS) and original geometry M1. Fig. 10(a) presents the maximum stress as function of scaling factor and speed where the maximum stress can be identified for different combination of the two parameters. The region depicted in yellow clearly shows mechanical unfeasible solutions which requires a further structural refinement stage. Fig. 10(b) presents the maximum displacement as function of scaling factor and speed.
Another mechanical consideration is related to the manufacturability of the rotor laminations. The thinnest part of the rotor lamination, i.e., the iron bridge, cannot be below a certain limit depending on the manufacturing method and selected material. In this case, it is not recommended to scale the original geometry M1 lower than S si < 0.75, as the ribs thickness will be lower than 0.45 mm.

VI. CASE STUDY AND EXPERIMENTAL VALIDATION
In this section, four geometries are evaluated and compared to the prototype M1 (geometry presented in Table I).    These were designed according to different scaling methods, two geometries using AGF (4)-(5) and two geometries using AGS (3), respectively. In the following subsections the evaluation of the torque ripple will be carried out for two current densities, 4 and 5 A/mm 2 , respectively, using the data from Table V. The winding configuration, is kept constant, whereas the number of turns per phase N s = 128 for all machines.

A. FE Torque Ripple Analysis for Scaled Machines
In Fig. 11(a) and (b) the results of the torque ripple analysis, conducted for reduced-scaled machines M4 and M5 (S s = 0.75), considering different current angles and loading, are shown. In Fig. 11(a) the ripple oscillations, evaluated for a current angle of 45 electrical degrees (α e = 45 o ), are presented. At J = 4 A/mm 2 their values are T ΔM4 = 15.1% and T ΔM5 = 13.23%, and at J = 5 A/mm 2 are T ΔM4 = 11.72% and T ΔM5 = 10.79%, for M4 and M5, respectively. It can be observed that M4 achieves higher torque for both current profiles compared to M5, this is mainly due to the increased air gap with respect to rotor size, when the AGF scaling is applied.
The same analysis has been carried out in a similar fashion for the scaled machines M2 and M3 (S s = 1.5). The FE simulation results are shown in Fig. 11(c) and (d). For a current density J = 4 A/mm 2 their values are T ΔM3 = 15.71% and T ΔM2 = 11.15%, while at J = 5 A/mm 2 are T ΔM3 = 16.7% and T ΔM2 = 11.69%, for M2 and M3, respectively. It can be observed that the torque ripple is increased for AGF scaled machine (M3), compared to AGS scaled (M2).
This confirms the behaviour shown in Fig. 4 and Fig. 6, where the machines that are scaled by AGF method have a significant ripple increase for machines with larger diameter. On the contrary, the average torque of the M3 is higher with respect to M2. For the sake of clarity, a summary of the above results is reported in Table VII, that will be described in the following section.

B. Experimental Results and Validation
In order to validate the proposed theory, the SynRel machine M1, with 24 slots 4 poles, has been tested on an instrumented rig. The stator and rotor laminations of the prototype are shown in Fig. 12.
The machine torque ripple has been characterised on a custom test rig presented in Fig. 13, described in detail in [26]. The tests are carried out at low speed in order to capture the high frequency nature of the torque oscillations. The motor M1 under test is connected through a torque meter to a master motor (dyno). Between the latter and the machine under test, a non-reversible gearbox is reducing the speed by a 1:59 ratio, as sketched in Fig. 13. The torque is measured for different current amplitudes and different current angles. The control algorithm is implemented on a dSpace 1104 platform.
At first the test was carried out at current angle α e = 45 o . Fig. 14(a) presents experimental and FE evaluation of the torque ripple waveforms at J = 4 A/mm 2 and J = 5 A/mm 2 , at α e = 45 o , respectively. As can be observed the torque ripple waveform determined via FE matches very well the experimental data. The measured torque ripple at J = 4 A/mm 2 is T ΔM1 = 13.4% with an average torque T = 0.56 Nm, whereas the FE evaluation gives T ΔM1 = 12.6% with average torque T = 0.576 Nm. For further validation, the same has been carried out for a higher current density value J = 5 A/mm 2 , where the measured torque ripple is T ΔM1 = 11.21% with average torque T = 0.89 Nm, whereas the FE evaluation gives a value of T ΔM1 = 11.13% with an average torque T = 0.89 Nm.
Based on these results, it can be said that the FE simulations predict the torque ripple accurately, with a slight under estimation. In fact, the average error of about δ FEA ∼ 2.4%. with respect to experimental data. Additional experimental tests have been carried out at a different current angle, α e = 50 o . Fig. 14(b) shown the experimental and FE evaluation of the torque ripple waveforms at J = 4 A/mm 2 and J = 5 A/mm 2 , at α e = 50 o , respectively.
Similarly, to Fig. 14(a), these results are confirming again that the FE simulated torque ripple waveforms are in line with the measured data. The experimentally obtained torque ripples for a current angle α e = 50 o , at J = 4 A/mm 2 , are T ΔM1 = 12.6% with average torque T = 0.567 Nm, whereas the FE ones T ΔM1 = 11.16% with average torque T = 0.577 Nm. For higher current density, J = 5 A/mm 2 , the experimental torque ripple is T ΔM1 = 11.21% with average torque T = 0.921 Nm, whereas FE T ΔM1 = 11.16% with average torque T = 0.926 Nm.
In order to summarise all the results from both experimental measurements and FE simulations, for the different operating conditions considered, Table VII is reporting the data for all machines analysed. Based on all the above it can be observed that the scaled machines present a torque ripple which is in close correlation with the reference machines, for both scaled and fixed air gap scaling principles. This is a confirmation that the scaling method can be used, starting from a machine optimised for a minimum torque ripple, to re-design a larger or smaller machine with minimum effort.

VII. CONCLUSION
This paper assesses the effect on the torque ripple of two homothetic scaling principles of synchronous reluctance machines. Two main scaling principles have been defined, which are the fixed air gap for the scaled machines AGF and scaled air gap for the scaled machines AGS. The correlations between the torque ripple of a reference machine with respect to a scaled machine is analysed in depth.
It has been demonstrated that the homothetic scaling method proposed leads to a design that can be considered optimal, or to a solution that is a good starting point for a further torque ripple optimization refinement. This approach is significantly reducing the computational time to obtain a machine with a minimum torque oscillation. In fact, all scaled machine has shown less than 5% increase in torque ripple with respect to reference machine. The torque ripple waveforms have been experimentally validated on manufactured prototype of the reference machine M1. The measured torque profiles are showing a very good match with respect to the FE evaluations. It can be concluded that the proposed method is defining a fast and accurate scaling technique for the preliminary design of the SynRel machines. This can be adopted by the industrial community, in particular when the performance assessment of a range of machine is required, starting from a reference design.