Smoothing the static torque characteristic of synchronous reluctance motor by Taguchi optimization method

Abstract

This paper focuses on finite element based modelling and optimization of synchronous reluctance machines (SynRM). The primary objective of the study was to improve the static torque characteristic of SynRM by appropriate selection of rotor geometry using design of experiments (DOE) by Taguchi optimization method. The preliminary finite element based numerical model of a SynRM was built and parametric analysis of rotor geometry was performed in order to investigate the impact of number of magnetic flux barriers and their position and size on the resulting static torque characteristic. The acquired characteristics were processed by Fourier analysis in order to identify the content of higher harmonics responsible for the machine torque ripple resulted by each rotor geometry. The processed results were used for smoothing the static torque characteristic of SynRM using DOE by Taguchi optimization method. The design of experiments optimization method by Taguchi was successfully used and implemented for the optimization of the SynRM rotor geometry in order to improve its static torque characteristic. Based on the obtained results the guidelines are proposed for smoothing the static torque characteristic by selecting the appropriate rotor geometry, which may in turn significantly contribute to the reduction of the torque ripple of SynRM.

Keywords

Synchronous reluctance motor numerical modelling finite element analysis optimization torque ripple Taguchi optimization method

1. Introduction

Synchronous reluctance motors (SynRM) present a promising alternative in a large variety of applications including automotive industry, traction systems, electricity driven technology by the ship industry and home appliances [1, 2, 3]. In the recent years, the SynRM is gaining considerable attention from both academia and industry due to many advantages they possess compared to other types of motors [2, 3]. First, the SynRM rotor design do not require conductive windings, which results in no copper losses from the rotor side, whilst the stator configuration is the same as the stator used in the state-of-the-art induction motor technology, which results in reduced manufacturing efforts. Second, the rotor can be constructed without the usage of permanent magnets, which contributes to the cost-effectiveness whilst there is no risk from the demagnetization due to a possible exposure to excessively high operation temperatures. In addition, the SynRM can be used also in harsh physical conditions to withstand high centrifugal forces and high temperatures. However, the torque ripple and the low power factor of SynRM still remain the major challenges in the electrical machine research and development [1, 2, 3, 4]. Namely, the torque ripple presents a particular inconvenient since it results in undesirable smooth-less rotor rotation transferred to the motor load, as well as in noise and mechanical vibrations emitted from the motor [4]. It has already been shown theoretically and experimentally that the torque ripple can be mitigated by selection of appropriate geometry configuration of the SynRM rotor and/or by appropriate current supply of the stator windings [5, 6]. The rotor geometry modification strategies resulting in reduced torque ripple involve proper relation between the number of stator slots per pole pair with respect to the number of the rotor segments (i.e. rotor flux barriers), rotor or stator skewing and geometrical modifications of the rotor design, as previously shown by our and other studies [5, 6]. Although, a large body of literature exits on ripple torque reduction, numerous studies are being conducted with the common goal of minimization and reduction of the ripple torque and its side effects to an acceptable level [1, 2, 3].

The primary objective of the present study was to improve the static torque characteristics of a four pole SynRM by appropriate selection of rotor geometry parameters by virtue of finite element method (FEM) numerical modelling and by design of experiments (DOE) optimization method by DOE Taguchi. The FEM numerical model of SynRM was built and parametric analysis of different rotor designs/geometry was performed using Ansys Maxwell software environment. The parametric analysis of different rotor designs was performed in order to investigate the impact of number of magnetic barriers, their size and position with respect to the shaft on the resulting static torque characteristic. The resulting torque characteristic were processed by Fourier analysis in order to identify the content of higher harmonics responsible for the machine torque ripple resulted by each rotor geometry we investigated. The results were implemented into the optimization process based on the design of experiments method (DOE) by Taguchi in order to specify the guidelines for reduction of torque ripple by selecting the appropriate rotor geometry.

2. Materials and methods

In the present paper we analyzed the static torque characteristics of a four pole synchronous reluctance machine (SynRM). The static torque characteristic is defined as relationship between the torque T generated by the machine and the load angle $\delta$ (i.e. also called function or signal T( $\delta$ )). We analyzed the SynRM rotor type with transversally laminated rotor with different numbers of magnetic flux barriers, which is frequently used in manufacturing process [1, 2]. The T( $\delta$ ) characteristics were analyzed for different geometrical parameters of SynRM rotors, while the stator configuration was kept identical for all investigated SynRM rotor configurations. The T( $\delta$ ) characteristics, also called signals, were studied by parametrization of rotor geometry in order to identify and minimize the content of the high harmonics responsible for the machine torque ripple by optimization of the machine rotor geometry.

Figure 1.

The illustration of the modelled SynRM configuration with the rotor containing one magnetic flux barrier.

The SynRM motor is driven by alternating current on the stator side and its rotor rotates in synchronism with the produced stator rotating magnetic flux density. The SynRM torque is produced based on the difference in reluctances in the d-axis and q-axis of the rotor (Fig. 1). The reluctances are here presented in form of inductances [6]. The SynRM torque is presented according to two-axis theory by the Eq. (1):

${T}=\frac{m\cdot p}{2}\cdot\frac{U^{2}}{\omega_{el}^{2}}\cdot\frac{(L_{d}-L_{q% })}{L_{d}\cdot L_{q}}\cdot\sin(2\delta),$ (1)

where $m$ is number of phases, $p$ is number of pole pairs, $U$ is the applied voltage, $\omega_{el}$ is electric frequency, $L_{d}$ is direct inductance (i.e. inductance in d-axis) and $L_{q}$ is the quadrature inductance (i.e. inductance in q-axis) and $\delta$ is the load angle.

The performance of the SynRM therefore strongly depends on the difference between the inductances in the d-axis and q-axis Eq. (1), which is defined by the geometry and material properties of the machine. In addition to the SynRM torque, the ratio between the two inductances determines also the power factor. The SynRM therefore needs to be designed with the highest possible difference and ratio between the $L_{d}$ and $L_{q}$ inductances in order to obtain the optimum performance [6].

According to the literature the quality of the SynRM torque characteristics strongly depends on rotor geometry [6]. Therefore, in order to find the optimum geometry which will result in an optimum torque characteristic T( $\delta$ ) we analyzed the influence of the following parameters of the SynRM rotor geometry: number of magnetic flux barriers, position of magnetic flux barriers with respect to the shaft (i.e. distance between the individual barrier and the shaft), and widths of each individual magnetic flux barrier. The parametric analysis of the listed geometric parameters was performed by virtue of the FEM numerical modelling applying non-linear magneto-static analysis using Ansys Maxwell software. The FEM analysis was chosen for the analysis in order to more accurately describe and calculate the torque characteristic resulting from the complex geometry of the rotor and stator, nonlinear material properties and non-sinusoidal excitation current distribution in the stator slots. In brief, our purpose was, to find a rotor geometry that would result in a static torque characteristic signal T( $\delta$ ) as close as possible to the one given by Eq. (1).

2.1 SynRM geometry and material properties definition

The geometry of the four pole SynRM stator and rotor is illustrated in Fig. 1. The figure shows the stator geometry with the arrangement of stator windings and the geometry of the one-barrier rotor design as an example of our models we built. The stator and rotor were separated with a 0.5 mm thick air gap. The ferromagnetic material M800-50A presented by BH characteristic was selected and applied for both the stator and rotor core layers. Similarly, for the material properties of the stator winding, the shaft and magnetic flux barrier the properties of iron and air were assigned from the software library. The three phases arranged within the stator slots (in this case nine slots per pole) are color coded as red, yellow and green for the U, V and W phases, respectively, as shown in Fig. 1.

2.2 Definition of stator geometry and windings configuration

The SynRM stator geometry was built according to the standardized guidelines for induction machine stator configurations from the Kienle Spiess catalogue library [7]. In this study we selected the stator geometry defined by the IEC 100/4.936 standard, which is typically used in 3 kW induction machines. According to the IEC 100/4.936 standard the stator geometry we built consisted of 36 slots; the outer stator diameter was 150 mm, the inner stator diameter was 90 mm, the area of the stator slot was 80 mm ${}^{2}$ and the length of the steel package was 110 mm. The resulting stator winding configuration consisted of a single winding layer with the number of turns per slot $N_{t}=$ 40 and the resulting fill factor $K_{cu}=$ 0.4. The windings were supplied by the three phase current system ( $i_{u}$ , $i_{v}$ and $i_{w}$ ) with the maximum current $I_{m}$ , supply frequency $\omega$ and phase shift $\varphi$ : $i_{u}=I_{m}$ *sin( $\omega t+\varphi$ ), $i_{v}=I_{m}$ *sin( $\omega t+2\pi/3+\varphi$ ) and $i_{w}=I_{m}$ *sin( $\omega t-2\pi/+\varphi$ ) for the phases $U$ , $V$ and $W$ , respectively. The magneto-static analysis was then carried out by the fixed time moment where the applied current of the phase $U$ was $I_{m}$ , and for the phases $V$ and $W$ were $-I_{m}$ /2. The magnitude of the supply current $I_{m}$ was 3.52 A, calculated from $I_{m}=$ J*( $\pi$ * $d_{Cu}^{2})$ /4, where the current density J for the air cooled motor was estimated to $J=$ 7 A/mm ${}^{2}$ [8], and the diameter of the cooper wire was $d_{cu}=$ 0.8 mm.

Figure 2.

Rotor geometry and definitions of geometry parameters and their levels (i.e. increments) for the studied rotor designs with: a) one, b) two and c) three magnetic flux barriers.

2.3 Parametric and optimization analysis of the rotor geometry

The investigated SynRM rotor geometries and corresponding parametrization definitions for the rotor configurations with different numbers of magnetic flux barriers is shown in Fig. 2. The rotor configuration with one barrier is defined with two parameters, where the parameter A denotes the distance of the magnetic flux barrier from the shaft and the parameter B is the width of the barrier, as shown in Fig. 2a. The rotor configurations with two and three magnetic flux barriers are shown in Fig. 2b and c, respectively. The two-barrier configuration is described with four geometrical parameters A, B, C and D (Fig. 2b), The parameters B and D indicate the widths of the who barriers, whereas the parameters A and C indicate the position of the barriers with respect to the shaft (i.e. distance between the barriers and the shaft). The three-barrier rotor configuration is described with six geometrical parameters A, B, C, D, E and G (Fig. 2c).

Table 1
The output design parameters obtained from DOE for one-barrier, two-barrier and three-barrier rotor designs

Design parameters		A	B	C	D	E	G
One-barrier	The best	L1	L1	$-$	$-$	$-$	$-$
	The worst	L3	L2	$-$	$-$	$-$	$-$
Two-barrier	The best	L3	L1	L3	L1	$-$	$-$
	The worst	L2	L3	L2	L3	$-$	$-$
Three-barrier	The best	L2	L1	L3	L1	L2	L1
	The worst	L1	L3	L2	L3	L1	L3

Each geometrical parameter is varied by the predefined increments, as described in Fig. 2, resulting in three different geometrical dimensions – from here on named levels. The geometry of SynRM rotor defined by each of the parameters was therefore investigated in three different levels (i.e. increments of geometrical dimensions) L1, L2 and L3, as shown in Fig. 2. Based on rotor geometry and parameter variation definitions described in Fig. 2 the parametric analysis was carried out in order to analyze the influence of rotor geometry modifications on the quality of the SynRM static torque characteristic. The parametric analysis is carried out based on the modification of the whole rotor geometry by automatic variation of A, B, C, D, E and G parameters in Ansys Maxwell. The automatic rotor geometry modification by parametric analysis requires definition of spatial coordinates of 30 points T(X,Y) describing the geometry of the three magnetic flux barrier, as shown in Fig. 3. The coordinates X and Y are therefore expressed as functions of A, B, C, D, E and G parameters. The equations we derived for the coordinates X and Y for the points T1 to T10 corresponding to the geometry of the first magnetic flux barrier are given in Appendix in Table 1. The equations for the coordinates of the points T11 to T20 corresponding to the geometry of the second magnetic flux barrier are given in Table 2, and the equations for the points T21 to T30 corresponding to the geometry of the third magnetic flux barrier are given in Table 3 of the Appendix.

The radius of the rotor shaft ( $R_{sh}=$ 14 mm), the radius of the magnetic flux ribs ( $R_{rib}=$ 44 mm) and the rotor radius ( $R_{rot}=$ 44.5 mm) were kept constant during the parametric analysis (Fig. 3).

Figure 3.

The position of the points from T1 to T30 describing the geometry of the first (T1to T10), second (T11 to T20) and third (T21 to T30) magnetic flux barrier. The radius of the rotor shaft, the radius of magnetic flux ribs and the rotor radius were Rsh $=$ 14 mm, Rrib $=$ 44 mm and Rrot $=$ 44.5 mm, respectively.

All numerical simulations were run on a personal computer platform with the 4-core processor with 2.7 GHz of CPU speed and 8GB of RAM and 128 GB of SSD disk. The accuracy of the numerical results was controlled by the selection of appropriate mesh density (i.e. the number of finite elements). The final mesh density was created by increasing the number of elements until the results of the calculated output results changed less than 0.5% and thus the numerical error was negligible.

In order to obtain the static torque characteristics T( $\delta$ ) of the studied rotor designs the load angle of the rotor was varied. The static torque characteristic was determined by rotating the rotor – increasing the load angle up to ninety degrees due to the symmetry of the four pole machine. The load angle was increased by increments of one degree in order to determine the static torque characteristics as precisely as possible. The interaction between stator slots/teeth and magnetic flux barriers in the rotor was then analyzed based on the determined torque dependency on the load angle.

The output of the parametric analysis was therefore the static torque characteristic T( $\delta$ ) for each studied rotor design (one, two, and three barrier configuration). Namely, the rotor with one barrier was described with two design parameters (A and B), the rotor with two-barriers was described with four parameters (A, B, C and D), while the rotor with three barriers was described with six parameters (A, B, C, D, E and G) and each of the design parameters was varied in three increments (also called levels L). The resulting number of T( $\delta$ ) characteristics is determined by the relationship: (number of levels) ${}^{\text{number of parameters}}$ . The final resulting number of T( $\delta$ ) for the SynRM with one barrier rotor design was 9, while for the SynRM with two barriers and three barriers rotor designs 81 and 729 T( $\delta$ ) characteristics, respectively were generated.

It is important to note that although good results can be obtained with the parametric analysis, it is in general time consuming process especially when a large number of parameters are required which is the case is in the present study. The process of the motor design described with the models involving a large number of parameters can be accelerated by using different optimization methods such as genetic algorithms or design of experiments provided that the appropriate optimization condition/constraint is defined and fulfilled [9]. In the present study the design of experiments (DOE) by Taguchi optimization method is used. The computation time needed for obtaining the optimum SynRM design for both the parametric analysis and the DOE optimization analysis is estimated and compared.

In order to evaluate the calculated T( $\delta$ ) characteristics the Fast Fourier Transformation (FFT) was applied. Ideally, the torque T( $\delta$ ) characteristic would have only the first harmonic (i.e. fundamental torque component). In reality, each SynRM design is influenced by torque ripple which is reflected in the high harmonics of the T( $\delta$ ) characteristic. Therefore, the purpose of the FFT was to analyze the quality of the static torque T( $\delta$ ) which consists of undesirable high harmonics. In this case, we analyzed the content of the first 20 high harmonics. The results of FFT analysis showed that the ninth harmonic was the highest and thus, the most critical in all designs (in pre optimization stage). The ninth harmonic was the most pronounced due to the stator geometry determined by 9 stator slots per pole. The results of FFT analysis served as input data for the optimization process performed with DOE by Taguchi method. The DOE optimization process was based on the following four scenarios defined according to the larger is better Taguchi design:

scenario 1 was defined to maximize the ratio between the peak value of the fundamental harmonic torque component and 9th high harmonic,

scenario 2 was defined to maximize the ratio between peak value of the fundamental harmonic torque component and the sum of the first 20 high harmonic torque components,

scenario 3 was defined to maximize the ratio between peak value of the fundamental harmonic torque component and the sum of three highest high harmonic torque components, and

scenario 4 was defined to maximize the ratio between the peak value of the fundamental harmonic torque component and the product of three highest peak values of torque high harmonics always obtained by FFT analysis.

It should be noted that by multiplying the three highest high harmonics the 4 ${}^{\rm th}$ scenario gives the importance of the minimum high harmonic component value, while the scenario 3 gives the importance to the maximum high harmonic component value.

The scenarios listed above were defined in order to identify the rotor geometry (i.e. to identify the optimum value of the parameters A, B, C, D, E and G) that will result in the T( $\delta$ ) signal with the highest fundamental torque component and with the minimum content of higher harmonic components responsible for the machine torque ripple.

The FFT was performed for all T( $\delta$ ) characteristics resulted from all investigated rotor designs. According to the defined scenarios, the Design of Experiments optimization by Taguchi method was carried out. The Design of Experiments by Taguchi method was performed by the software Minitab 17.

Figure 4.

The optimum results obtained with the DOE by Taguchi optimization method taking into account the defined four optimization scenarios for the SynRM configuration with: a) one magnetic flux barrier, b) two magnetic flux barriers and c) three magnetic flux barriers.

3. Results and discussion

The optimum results obtained with the DOE by Taguchi optimization method taking into account the defined four optimization scenarios are shown in Fig. 4. The highest ratios defined by each optimization scenario (maximum values marked with red circles in Fig. 4) indicate the best/optimum rotor designs calculated by DOE optimization for rotor designs with one (Fig. 4a), two (Fig. 4b) and three magnetic flux barriers (Fig. 4c). The lowest ratio defined by each optimization scenario indicates the worst rotor design (i.e. minimum values displayed in Fig. 4) resulting in the lowest quality of static torque characteristics.

Figure 5.

Distribution of magnetic flux density B(T) within the SynRM configurations - comparison between the best (left panel) and the worst rotor designs (right panel) obtained by DOE optimization method.

For the rotor with one barrier the DOE optimization selected and analyzed all nine T( $\delta$ ) characteristics, while for the rotors with two-barriers the DOE selected 9 out of 81 and for the three-barriers 27 out of the total 729 characteristics. The Taguchi orthogonal arrays for the rotors with one, two and three barriers are given in Tables 4 and 6, respectively (see the Appendix). The reduction of total optimized characteristics significantly reduced the computational time, which is particularly useful for the models with large number of optimization parameters (such as in our case the model of SynRM with multi-barrier rotor designs). Namely, the computation time needed for the model with one set of parameters and for one single position of the rotor with respect to the stator was 7 seconds (i.e. for the increment of 1 ${}^{\circ}$ ). The computation time needed for calculation of one T( $\delta$ ) characteristic was 630 seconds (i.e. 10.5 min). The computation time for the entire parametric analysis of the SynRM with the three barriers with 729 possible combinations was therefore 7654.5 min. The computation time needed for the optimization analysis by DOE for the SynRM with the three barriers with for the 27 combinations out of 729 was 278.1 min (4.635 h).

From the output results obtained by DOE the best and the worst rotor designs were determined based on the scenarios which resulted in equal output values of geometric parameters (Fig. 4). Therefore, the design parameters for the rotor with one magnetic flux barrier were selected based on the scenarios 2, 3 and 4, since the three scenarios gave the same optimization output results. For the rotor with two-magnetic flux barriers the design parameters were selected based on scenarios 2 and 3. For the rotor with three-magnetic flux barriers the design parameters were selected based on scenarios 1, 2, 3 and 4 (i.e. all scenarios resulted in the same output results, Fig. 4). The output results (i.e. optimum/best versus the worst design parameters) obtained from DOE for each of the investigated SynRM configurations are summarized in Table 1.

The comparison of calculated magnetic flux density distribution within the investigated SynRM configurations with rotors designs with one-barrier, two-barriers and three barriers is given in Fig. 5. The best designs obtained with DOE optimization of geometrical parameters for each rotor design are displayed in the left panel, while the worst designs are displayed in the right panels. From the left panel of the Fig. 5 (the zoomed view) it can be seen that the magnetic flux density $B$ in the rotor area does not exceed the saturation level of the material (i.e. the saturation point of the material M800-50A which is used in this study is $B=$ 1.72 T). However, the value higher above 1.72 T up to 2.5 T appear in the small area where the bridges (to increase the mechanical strength of rotor lamination) saturates. The Fig. 5d and f show that the $B$ in the rotor lamination of the worst designs exceeds the saturation level (see the zoomed area in the right panel).

Figure 6.

Magnetic flux density B(T) distribution in the air gap (comparison between the best (bold dashed line) and the worst rotor design (solid line)) with: a) one, b) two and c) three magnetic flux barriers.

For the SynRM investigated in this study, the results displayed in Fig. 5a, c and e indicate that the optimum rotor design obtained by DOE is when the magnetic flux barriers widths are as narrow as possible for all investigated rotor configurations (as indicated by the optimum values of the design parameters determined by DOE – Table 1). The one-barrier design is at its optimum (Fig. 5a) when the magnetic flux barrier is as close as possible to the shaft (i.e. L1 for the design parameter A – Table 1), while the optimum for the two-barrier design (Fig. 5c) is obtained when the magnetic flux barriers are positioned as far as possible from the shaft (i.e. L3 for the design parameters A and C – Table 1). The optimum positioning of barriers for the rotor with three-barriers (Fig. 5e) is when the central barrier is positioned as far as possible from the shaft (i.e. L3 for the design parameter C – Table 1), while the inner and outer barriers are positioned in the middle of the optimization limits (i.e. L2 for the design parameters A and E – Table 1).

In order to better understand the magnetic interaction between the stator and rotor the acquisition of the magnetic flux density in the air gap was performed. The comparison of magnetic flux density distribution in the air gap for the best (dashed line) and the worst (solid line) rotor design is given below in Fig. 6. The magnetic flux density B is displayed along the central arc in the air gap along the $l_{a}$ , as shown in Fig. 3 ( $la=$ 2*pi*R/4 for the $R=$ 44.75 mm).

The results $B(l_{a})$ shown in Fig. 6 indicate that the magnetic field density in the air gap is in a strong correlation with the rotor geometry/configuration. The geometry configuration with nine slots per pole is reflected in a sequence of nine peaks of magnetic field density in the air gap. In the best design the resulting signal drawn over the nine individual peak signals represent a smooth sinusoid curve. On the other hand, the resulting signal drawn over the nine peaks of the worst design significantly deviates from the sinusoid curve (due to the stronger influence of higher harmonics). Namely, in the poor designs the presence of high harmonics in the magnetic field density is more pronounced, hence the B( $l a$ ) is not smooth which is reflected in the shape of the torque signal.

Figure 7.

The calculated static torque characteristics T( $\delta$ ) and the FFT analysis of the T( $\delta$ ) characteristics resulting from the most optimal rotor designs with: one, two and three magnetic flux barriers.

The calculated static torque characteristics T( $\delta$ ) (left panel) and the FFT analysis of the T( $\delta$ ) (right panel) characteristics resulting from the most optimal rotor designs obtained with DOE optimization procedure for: one, two and three magnetic flux barriers are shown in Fig. 7. The calculated static torque characteristics T( $\delta$ ) (left panel) and the FFT analysis of the T( $\delta$ ) (right panel) characteristics resulting from the least optimal rotor design obtained as output result from DOE optimization procedure for: one, two and three magnetic flux barriers are shown in Fig. 8.

Figure 8.

The calculated static torque characteristics T( $\delta)$ and the FFT analysis of the T( $\delta)$ characteristics resulting from the least optimal (worst) rotor designs with: one, two and three magnetic flux barriers.

In order to determine the most influential harmonics in all studied designs (i.e. with one, two and three magnetic flux barriers) the mean value of the torque was calculated for each harmonic. The results of the worst rotor designs show that the 9th harmonic was, as expected, the highest and thus the most pronounced higher harmonic, which is attributed to the nine stator slots per pole. In the SynRM with one magnetic flux barrier the second and the third most critical harmonics were are the 8th and the 10 ${}^{\rm th}$ (Fig. 8). In the worst design with two barriers the second and third most critical harmonics were the 3rd and the 7th, while in the design with three barriers the second and third most critical harmonics were the 2nd and the 8 ${}^{\rm th}$ ones (as shown in Fig. 8).

The comparison of the results in Figs 7 and 8 demonstrates that the higher harmonics responsible for the torque ripple are significantly reduced which results in a significantly smoother static torque characteristic obtained with the best/the most optimal rotor configurations. The difference in the resulting static torque signals obtained with the one-barrier, two-barrier and three-barrier rotor designs occurs due to the different maximum value of the fundamental harmonic torque component and different content of the most influential high harmonics, being directly correlated to the rotor geometry.

The two-barrier design results in a better torque characteristic (i.e. in terms of higher value of the fundamental torque component and lower presence of higher harmonics) compared to the one-barrier design. Namely, the fundamental torque value of the two-barrier design was by 10% higher compared to maximum value of the fundamental torque component obtained with the one-barrier design. The rotor design with three magnetic barriers resulted in the lowest value of the fundamental torque component compared to the one and two-barrier designs. Namely, the three-barrier design resulted in a 15% and 25% lower value of the fundamental torque component compared to the value of the fundamental torque component of the one and two-barrier designs, respectively. However, it has the lowest content of the higher harmonics of the static torque signal, which could be attractive for sensitive drives where the minimum possible ripple is required.

To summarize, when comparing the three SynRM configurations, the rotor with two magnetic flux barriers resulted in the most optimal static torque characteristic in terms of the maximum value of the fundamental torque component with minimum content of the high harmonic torque components. The three-barrier rotor design resulted in in the lowest content of the high harmonics that are responsible for the torque ripple, however the value of the fundamental torque component is smaller in comparison with the rotor with two barrier.

Based on the modelling and optimization of one, two and three barrier rotor designs the following equation can be also derived:

$\frac{N_{ss}}{p\cdot N_{\textit{MFB}}}\neq\text{intiger},$

where $N_{ss}$ is the number of the stator slots, $p$ is the number of the rotor poles and the $N_{\textit{MFB}}$ is the number of the magnetic flux barriers.

It can therefore be concluded that the ratio between the number of stator slots and the product of the number of rotor poles and the number of the magnetic flux barriers should not be an integer in order to obtain the static torque characteristic as smooth as possible with the maximum value of the fundamental torque component (i.e. the static torque characteristic with both the maximum value of the fundamental torque component and the lowest content of the high harmonics). This conclusion can be employed also to the other types of the SynRM with different numbers of stator slots.

4. Conclusions

In this paper the static torque characteristic of the SynRM was analyzed and evaluated for different rotor configurations by using parametric analysis of rotor geometry and DOE optimization method by Taguchi. Different rotor geometries (one, two, and three barrier configuration) and their influences on the static torque characteristic were studied and compared. The output results we obtained from all studied rotor designs show that the magnetic flux barrier position with respect to the shaft and magnetic flux barrier’s width have a strong impact on the static torque characteristic of the SynRM.

Based on the SynRM modelling and optimization output results it can be concluded that the ratio between the number of stator slots and the product of the number of rotor poles and the number of the magnetic flux barriers should not be an integer in order to obtain a static torque characteristic as smooth as possible with the maximum value of the fundamental torque component. This conclusion can be employed also to the other SynRM configurations with different number of stator slots.

The results of this study can be applied for optimization, design and manufacturing process of SynRM machines with improved static torque characteristic and thus reduced torque ripple of the machine. The findings of this study can also be useful for further advanced theoretical and experimental analysis of SynRM machines including hybrid SynRM machines such as for example permanent magnet assisted SynRM.

Footnotes

Acknowledgments

This work is supported by the Slovenian Research Agency (project ID L2-8187 (C)). The authors acknowledge the project entitled Premium efficiency class electrical motors (ID L2-8187 (C)) was financially supported by the Slovenian Research Agency.

Appendix

Table 1

Definition of X and Y coordinates of points $T_{1}$ to $T_{10}$ for geometric parametrization of the first magnetic flux barrier

Point	X-coordinate	Y-coordinate
$T_{1}$	$\sqrt{2}*(R_{sh}+A)-A$	A
$T_{2}$	A	$\sqrt{2}*(R_{sh}+A)-A$
$T_{3}$	$\sqrt{R_{rib}^{2}-A^{2}}$	A
$T_{4}$	A	$\sqrt{R_{rib}^{2}-A^{2}}$
$T_{5}$	$\sqrt{R_{rib}^{2}-\left(A+\frac{B}{2}\right)^{2}}$	$A+\frac{B}{2}$
$T_{6}$	$A+\frac{B}{2}$	$\sqrt{R_{rib}^{2}-\left(A+\frac{B}{2}\right)^{2}}$
$T_{7}$	$\sqrt{2}*\left(R_{sh}+A+B\right)-A-B$	${A+B}$
$T_{8}$	$A+B$	$\sqrt{2}*\left(R_{sh}+A+B\right)-A-B$
$T_{9}$	$\sqrt{R_{rib}^{2}-(A+B)^{2}}$	$A+B$
$T_{10}$	$A+B$	$\sqrt{R_{rib}^{2}-(A+B)^{2}}$

Table 2

Definition of X and Y coordinates of points $T_{11}$ to $T_{20}$ for geometric parametrization of the second magnetic flux barrier

Point	X-coordinate	Y-coordinate
$T_{11}$	$\sqrt{2}*\left(R_{sh}+C\right)-C$	C
$T_{12}$	C	$\sqrt{2}*\left(R_{sh}+C\right)-C$
$T_{13}$	$\sqrt{R_{rib}^{2}-C^{2}}$	C
$T_{14}$	C	$\sqrt{R_{rib}^{2}-C^{2}}$
$T_{15}$	$\sqrt{R_{rib}^{2}-\left(C+\frac{D}{2}\right)^{2}}$	$C+\frac{D}{2}$
$T_{16}$	$C+\frac{D}{2}$	$\sqrt{R_{rib}^{2}-\left(C+\frac{D}{2}\right)^{2}}$
$T_{17}$	$\sqrt{2}*\left(R_{sh}+C+D\right)-C-D$	$C+D$
$T_{18}$	$C+D$	$\sqrt{2}*\left(R_{sh}+C+D\right)-C-D$
$T_{19}$	$\sqrt{R_{rib}^{2}-{(C+D)}^{2}}$	$C+D$
$T_{20}$	$C+D$	$\sqrt{R_{rib}^{2}-{(C+D)}^{2}}$

Table 3

Definition of X and Y coordinates of points $T_{21}$ to $T_{30}$ for geometric parametrization of the third magnetic flux barrier

Point	X-coordinate	Y-coordinate
$T_{21}$	$\sqrt{2}*\left(R_{sh}+E\right)-G$	E
$T_{22}$	E	$\sqrt{2}*\left(R_{sh}+E\right)-E$
$T_{23}$	$\sqrt{R_{rib}^{2}-E^{2}}$	E
$T_{24}$	E	$\sqrt{R_{rib}^{2}-E^{2}}$

Table 3, continued
Point	X-coordinate	Y-coordinate
$T_{25}$	$\sqrt{R_{rib}^{2}-\left(E+\frac{G}{2}\right)^{2}}$	$E+\frac{G}{2}$
$T_{26}$	$E+\frac{G}{2}$	$\sqrt{R_{rib}^{2}-\left(E+\frac{G}{2}\right)^{2}}$
$T_{27}$	$\sqrt{2}*\left(R_{sh}{+E+G}\right){-E-G}$	${E+G}$
$T_{28}$	${E+G}$	$\sqrt{2}*\left(R_{sh}{+E+G}\right){-E-G}$
$T_{29}$	$\sqrt{R_{rib}^{2}-{{(E+G)}}^{2}}$	${E+G}$
$T_{30}$	${E+G}$	$\sqrt{R_{rib}^{2}-{{(E+G)}}^{2}}$

Table 4

Taguchi orthogonal array table for the model of the SynRM rotor with one magnetic flux barrier

Experiment number	A	B
1	L1	L1
2	L1	L2
3	L1	L3
4	L2	L1
5	L2	L2
6	L2	L3
7	L3	L1
8	L3	L2
9	L3	L3

Table 5

Taguchi orthogonal array table for the model of the SynRM rotor with two magnetic flux barriers

Experiment number	A	B	C	D
1	L1	L1	L1	L1
2	L1	L2	L2	L2
3	L1	L3	L3	L3
4	L2	L1	L2	L3
5	L2	L2	L3	L1
6	L2	L3	L1	L2
7	L3	L1	L3	L2
8	L3	L2	L1	L3
9	L3	L3	L2	L1

Table 6

Taguchi orthogonal array table for the model of the SynRM rotor with three magnetic flux barriers

Experiment number	A	B	C	D	E	G
1	L1	L1	L1	L1	L1	L1
2	L1	L1	L1	L1	L2	L2
3	L1	L1	L1	L1	L3	L3
4	L1	L2	L2	L2	L1	L1
5	L1	L2	L2	L2	L2	L2
6	L1	L2	L2	L2	L3	L3
7	L1	L3	L3	L3	L1	L1
8	L1	L3	L3	L3	L2	L2
9	L1	L3	L3	L3	L3	L3
10	L2	L1	L2	L3	L1	L2
11	L2	L1	L2	L3	L2	L3
12	L2	L1	L2	L3	L3	L1

Table 6, continued
Experiment number	A	B	C	D	E	G
13	L2	L2	L3	L1	L1	L2
14	L2	L2	L3	L1	L2	L3
15	L2	L2	L3	L1	L3	L1
16	L2	L3	L1	L2	L1	L2
17	L2	L3	L1	L2	L2	L3
18	L2	L3	L1	L2	L3	L1
19	L3	L1	L3	L2	L1	L3
20	L3	L1	L3	L2	L2	L1
21	L3	L1	L3	L2	L3	L2
22	L3	L2	L1	L3	L1	L3
23	L3	L2	L1	L3	L2	L1
24	L3	L2	L1	L3	L3	L2
25	L3	L3	L2	L1	L1	L3
26	L3	L3	L2	L1	L2	L1
27	L3	L3	L2	L1	L3	L2

References

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I.C.

Spyropoulos

D.V.

and Mitronikas

E.D.

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Yilmaz

, Limitations/capabilities of electric machine technologies and modelling approaches for electric motor design and analysis in plug-in electric vehicle applications, Renewable and Sustainable Energy Reviews 52 (2015), 80–99, doi: 10.1016/j.rser.2015.07.033.

Taghavi

and Pillay

, A sizing methodology of the synchronous reluctance motor for traction applications, IEEE Journal of Emerging and Selected Topics in Power Electronics 2(2) (2014), 329–340, doi: 10.1109/JESTPE.2014.2299235.

Ibrahim

M.N.

Sergeant

and Rashad

E.M.

, Synchronous reluctance motor performance based on different electrical steel grades, IEEE Transaction on Magnetics 51(11) (2015), 7403304, doi: 10.1109/TMAG.2015.2441772.

Vagati

Pastorelli

Franceschini

and Petrache

S.C.

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Miljavec

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Marčič

Štumberger

and Zagradišnik

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C.C.

Lyu

L.Y.

Liu

C.T.

and Li

P.L.

, Optimal design of an SPM motor using genetic algorithms and taguchi method, IEEE Transactions on Magnetics 44(11) (2008), 4325–4328, doi: 10.1109/TMAG.2008.2001526.

Experiment number	A	B	C	D
1	L1	L1	L1	L1
2	L1	L2	L2	L2
3	L1	L3	L3	L3
4	L2	L1	L2	L3
5	L2	L2	L3	L1
6	L2	L3	L1	L2
7	L3	L1	L3	L2
8	L3	L2	L1	L3
9	L3	L3	L2	L1

Experiment number	A	B	C	D	E	G
1	L1	L1	L1	L1	L1	L1
2	L1	L1	L1	L1	L2	L2
3	L1	L1	L1	L1	L3	L3
4	L1	L2	L2	L2	L1	L1
5	L1	L2	L2	L2	L2	L2
6	L1	L2	L2	L2	L3	L3
7	L1	L3	L3	L3	L1	L1
8	L1	L3	L3	L3	L2	L2
9	L1	L3	L3	L3	L3	L3
10	L2	L1	L2	L3	L1	L2
11	L2	L1	L2	L3	L2	L3
12	L2	L1	L2	L3	L3	L1

Experiment number	A	B	C	D
1	L1	L1	L1	L1
2	L1	L2	L2	L2
3	L1	L3	L3	L3
4	L2	L1	L2	L3
5	L2	L2	L3	L1
6	L2	L3	L1	L2
7	L3	L1	L3	L2
8	L3	L2	L1	L3
9	L3	L3	L2	L1

Experiment number	A	B	C	D	E	G
1	L1	L1	L1	L1	L1	L1
2	L1	L1	L1	L1	L2	L2
3	L1	L1	L1	L1	L3	L3
4	L1	L2	L2	L2	L1	L1
5	L1	L2	L2	L2	L2	L2
6	L1	L2	L2	L2	L3	L3
7	L1	L3	L3	L3	L1	L1
8	L1	L3	L3	L3	L2	L2
9	L1	L3	L3	L3	L3	L3
10	L2	L1	L2	L3	L1	L2
11	L2	L1	L2	L3	L2	L3
12	L2	L1	L2	L3	L3	L1