Magnetic Equivalent Circuit and Optimization Method of a Synchronous Reluctance Motor with Concentrated Windings

Abstract

In this paper, the 12-slot/4-pole (12/4) synchronous reluctance motor (SynRM) with concentrated windings is proposed for low-cost hybrid vehicles. The non-linear magnetic equivalent circuit (MEC) model of the 12/4 SynRM is built to obtain the main electromagnetic characteristics such as coil flux, inductances, torque, etc. The magnetic saturation is also counted in by the MEC. Results calculated by MEC are validated by 2D finite element analysis (FEA). Then, aiming at larger average torque, lower torque ripple and lower total harmonic distortion (THD) in phase voltage, the parameter optimization method of the SynRM is proposed based on the Taguchi method and the MEC model. The proposed Taguchi-MEC method enables a fast optimization with satisfactory accuracy. Finally, the motor prototype is manufactured, and experimental validations are carried out.

1. Introduction

Thanks to the simple structure and robust rotor without a brush and slip ring, the synchronous reluctance motor (SynRM) has become an attractive candidate [1,2,3]. Conventional SynRMs employ distributed windings to obtain essentially sinusoidal back electromotive force (back-EMF) per phase and a high winding factor [4,5,6,7]. However, due to the long end-part winding length, distributed windings result in increased copper loss, decreased efficiency and power density. On the other hand, concentrated windings are rarely employed for SynRMs because of the high harmonic distortion in armature windings and low winding factor. Spargo and other authors introduce fractional slot concentrated windings (FSCW) to SynRM, with a segment stator to improve torque density and efficiency [8,9,10,11,12,13,14], e.g., the 6-slot 4-pole (6/4) SynRel machine. The 6/4 machine exhibits a comparable performance of a permanent magnet (PM)-assisted SynRM with a high torque output capability and a low level of torque ripple [8,9]. The proposed 12/4 CW machine exhibits comparable torque performance, compared to the case of adopting distributed winding, when the total stack length is unchanged. The short winding end, PM-free and very low mutual inductances of the 12/4 CW machine will be appreciated by applications where a short stack length, high robustness and low cost are required, such as the integrated starter generator (ISG) in low-cost wild hybrid vehicles. The ISG is located between the engine and the gear box, and drives the output shaft directly. First of all, the vehicle producer requires a brushless and PM-less motor and concentrated windings. Getting rid of permanent magnets reduces the cost and enhances robustness and speed range. Avoiding distributed windings improves the air flow and makes it more dustproof, as well as other mechanical considerations. These above requirements leave only the choices of the reluctance motor and SynRM. The 9/6 or 15/8 SynRM are not considered due to unbalanced radial forces on the rotor. More importantly, low mutual inductances between armature phases is highly appreciated, since it will facilitate a high fault-tolerant capability, which can be found in the 12/4 SynRM machine.

Consequently, the magnetic equivalent circuit (MEC) model is built to predict electromagnetic performance, including air-gap flux density distributions, winding inductances and electromagnetic torque. Then, the obtained results are validated by 2D-finite element analysis (2D-FEA). Furthermore, aiming at better torque performances and lower total harmonic distortion (THD) in phase voltage, the optimization of the 12/4 SynRM is carried out combing the MEC and Taguchi method. Unlike traditional optimization methods using FEA coupling optimization algorithms, such as a genetic algorithm [15,16,17,18] and particle swarm algorithm [19,20,21,22], the Taguchi method exhibits the merit of being less time consuming [23,24,25,26]. This merit can be greatly enhanced by using MEC instead of FEA, maintaining satisfactory accuracy.

2. Machine Topologies and Design Specifications

This section provides a brief introduction to a prototype of the 12/4 SynRM. The topology structure of the designed 12/4 SynRM is shown in Figure 1, with the key dimensional parameters listed in

Table 1. Main Design Specifications of the 12/4 Machine.

Item	Value
Outer diameter	150 mm
Lamination length	75 mm
Total end winding	20 mm
Total stack length	95 mm
β2	12.4°
wb1	10.00 mm
stooth	11.12 mm
syoke	11.38 mm
β1	11.10°
dm1	4.41 mm
wrib1-1	0.58 mm
wrib2	0.28 mm
dm2	3.50 mm
wrib1-2	0.50 mm
lso	1.0 mm
Turns per coil	30
Copper fill factor	0.58
d-axis inductance (FEA)	19.3 mH
q-axis inductance (FEA)	6.2 mH
Phase current	30.1 A
Current angle	55°

. The outer dimensions of the stator are chosen based on the available test bench and power converters for the following experiment. The segmented stator core is designed to improve the slot filling factor (SFF). Torque ripple reduction also needs to be mitigated, so that a continuous skew rotor is adopted. It should be noted that the skewing angle is optimized to achieve a balance between the sinusoidal back-EMF and electromagnetic torque. Two rotors are manufactured, one without skewing and another with skewing. This paper only introduces the 12/4 SynRM without rotor skewing. The skewed rotor will be introduced in another coming paper. Moreover, the seam/gap between the neighboring stator segments depends on the manufacturing accuracy and are normally less than 0.05 mm. Considering these seams are very small, their influences on magnetic performances can be neglected and not considered in the MEC and FEA models.

3. Magnetic Equivalent Circuit Method and Evaluation

As an efficient tool, the MEC method has been widely utilized to analyze the electromagnetic performance of electrical machines [27,28], and is considered as a compromise between FEA with high accuracy and an analytical method (AM) with straightforward physical connections of performance and geometrical parameters. Therefore, the MEC model of the 12/4 SynRM is built in this paper. Figure 2 illustrates a quarter of the 12/4 machine, where each specific magnetic path is illustrated by a different MEC branch. The SynRM is separated into segments according to the geometrical shape. The equivalent permeance of each part of the stator and rotor core can be calculated using the permeance equations.

The basic equation which governs each branch of the magnetic circuit is given by,

$$\frac{\Phi }{G}=F$$

(1)

$$G={\mu }_i{\mu }_0\frac{\mathrm{S}}{l}$$

(2)

In Equation (1), Φ, G and F are the flux, permeance and magnetic-motive-force (MMF), respectively. In Equation (2), μ_r, μ₀, S and l are the relative permeability, the permeability of free space, the cross-section area and the length, respectively. Specifically, μ_r is determined by iteration from the B–H curve of the lamination material. The permanent magnets can simply be modeled as an equivalent MMF [29],

$$F_m=\frac{B_r}{{\mu }_i{\mu }_0}{h}_m$$

(3)

and permeance,

$$G_m={\mu }_i{\mu }_0\frac{l_m{l}_a}{h_m}$$

(4)

where h_m and l_m are the magnet thickness and width, respectively, and B_r is the remanence.

Then, the MEC model is solved by MATLAB following the flow chart shown in Figure 3. The saturation in iron is counted in by the iteration of μ_r. For each branch in the magnetic circuit at the ith step in an iteration, the condition of convergence ε is defined as,

$$\epsilon =\left|\frac{{\mu }_i-{\mu }_i^{\prime }}{{\mu }_i}\right|\le {10}^{-6}$$

(5)

where μ_i is the permeance given before equation solving, and μ’_i is the real permeance obtained after the equation solving, as shown in Figure 4. If ε ≤ 10⁻⁶ is not achieved, the permeance at the (i + 1)th step in an iteration is given by Equation (6) and k = 0.5 in this case.

$${\mu }_{i+1}=\left(1-k\right){\mu }_i+k{\mu }_i^{\prime }, 0<k<1$$

(6)

Figure 5 shows the mesh results of the FEA model in which the air-gap consists of 6 layers of elements. The path to plot the air-gap flux density distribution is also given in Figure 5a, which covers 1/4 of the air-gap. The air-gap flux density distributions generated by phase-A current, d-axis current and q-axis current, respectively, are predicted by MEC and then compared to the FEA results by Figure 6. Overall, satisfied agreements are achieved.

4. Results and Discussion

When the MEC model of the 12/4 SynRM motor is solved, the electromagnetic characteristics can be easily calculated, e.g., the flux in the air-gap (Φ_A), coil flux linkage (Ψ_a), phase-A self-inductance (L_aa) and mutual inductance between phase-A and -B (M_ba) can be obtained by

$${\Phi }_A=\int BdS$$

(7)

$${\Psi }_A=N\left({\Phi }_A+{\Phi }_{\delta A}\right)$$

(8)

$$L_{aa}=\frac{{\Psi }_A}{I_A}$$

(9)

$$M_{ba}=\frac{{\Psi }_B}{I_A}$$

(10)

Then, the L_d (d-axis), L_q (q-axis) and L_dq (d-q coupled inductance) can be calculated by Equations (11) and (12). Figure 7 compares the MEC and FEA obtained inductances with good agreements.

$$\left[\begin{array}{c}{L}_d\\ L_q\\ L_{dq}\end{array}\right]=P\left[\begin{array}{ccc}{L}_{aa}& M_{ba}& M_{ca}\\ M_{ab}& L_{bb}& M_{cb}\\ M_{ac}& M_{bc}& L_{cc}\end{array}\right]P^{-1}$$

(11)

$$P=\frac{2}{3}\left[\begin{array}{ccc}\cos \theta & \cos \left(\theta -\frac{2}{3}\pi \right)& \cos \left(\theta +\frac{2}{3}\pi \right)\\ -\sin \theta & -\sin \left(\theta -\frac{2}{3}\pi \right)& -\sin \left(\theta +\frac{2}{3}\pi \right)\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right]$$

(12)

$$P^{-1}=\left[\begin{array}{ccc}\cos \theta & -\sin \theta & \frac{1}{2}\\ \cos \left(\theta -\frac{2}{3}\pi \right)& -\sin \left(\theta -\frac{2}{3}\pi \right)& \frac{1}{2}\\ \cos \left(\theta +\frac{2}{3}\pi \right)& -\sin \left(\theta +\frac{2}{3}\pi \right)& \frac{1}{2}\end{array}\right]$$

(13)

It should be noted that in the 12/4 SynRM with concentrated windings, there is a cross-coupling between the d-axis and q-axis, i.e., the L_dq contributes unneglectable electromagnetic torque, as given in Equation (14).

$$T=\frac{3}{2}p{i}_s{}^2\left[\frac{1}{2}\left(L_d-L_q\right)\sin 2\alpha +L_{dq}\cos 2\alpha \right]$$

(14)

where p is the rotor pole-pairs, L_d and L_q is the d- and q-axis inductance, respectively, and α is the current phase angle, i.e., the angle between the current vector and d-axis. Thus, the torque waveform can be calculated. Figure 8 and Figure 9 give the torque performance versus current phases and rotor position, respectively. As can be seen, good agreements are achieved between the MEC and FEA results.

5. Optimization of Torque Characters Based on MEC and Taguchi Method

The Taguchi method is chosen as one of the many optimization methods. The optimization methods introduced in Section 1 require sophisticated algorithms and additional programming and data processing along with a lot of FEA cases of electromagnetic field. So, the methods are usually time consuming and complex. On the other hand, the Taguchi method can obtain satisfactory results by reduced calculation cases, and thus is less time consuming. In this paper, the Taguchi method and the MEC model are used to optimize the SynRM. The steps in the Taguchi method are as follows [30]:

• Select the parameters and determine its levels.
• Design of exams.
• Review the optimization results.

First of all, the means of all results M_t(S) and average effect M_Xj can be calculated as

$$M_t\left(S\right)=\frac{1}{n}\sum _{p_n=1}^n{S}_{p_n}$$

(15)

$$M_{X_j}\left(Y\right)=\frac{1}{3}\left(Y_1+Y_2+Y_3\right)$$

(16)

where X_i and Y_i are factor-level and performances, respectively. Then, the analysis of variance is carried out, which provides a measure of confidence. The technique does not directly analyze the data, but rather determines the variance of the data. The sum of squares (SS) is calculated first. It is a measure of the deviation of the experimental data from the mean value of the data. The sum of squares due to various factors can be calculated as

$$SS=j\sum _{p_j=1}^j{\left(M_{X_{p_j}}\left(Y\right)-M_t\left(S\right)\right)}^2$$

(17)

where M_t(S) and M_xpj(Y) are results of Equations (5) and (6). SS shows the relative importance of various factors on machine performance. Then, the Pearson coefficient is chosen to calculate the coefficient of correlation and given as

$${\rho }_{X_i,Y_i}=\frac{n\sum X_i{Y}_i-\sum X_i\sum Y_i}{\sqrt{n\sum X_i^2-{\left(\sum X_i\right)}^2}\sqrt{n\sum Y_i^2-{\left(\sum Y_i\right)}^2}}$$

(18)

The optimization aim is a larger average torque (T_avg), lower torque ripple (T_rip) and lower THD of phase voltage (THD_U). Firstly, the optimization parameters are determined and shown in Figure 10 and listed in

Table 2. Design Parameters and Their Variation Levels.

Parameter	Level 1	Level 2	Level 3
β1	11.5°	12.5°	12°
β2	11.5°	12.5°	12°
dm1	3.5 mm	4.5 mm	4 mm
dm2	2.5 mm	3.5 mm	3 mm
wb1	3 mm	4 mm	3.5 mm
wrib1	0.55 mm	0.65 mm	0.6 mm
wrib2	0.45 mm	0.55 mm	0.5 mm
syoke	10 mm	12 mm	11 mm
stooth	10 mm	12 mm	11 mm
lso	0.8 mm	1.2 mm	1 mm

, which are related to the rotor barriers and ribs, stator tooth and yoke width.

The design of the exam is carried out after preparing the initial information. To obtain the number of exams, the orthogonal arrays are presented using Taguchi. The exams’ matrix can be easily created using existing software, in this case the Minitab. Shown in

Table 3. Exams Table.

Exams	A	B	C	D	E	F	G	H	J	K
1	1	1	1	1	1	1	1	1	1	1
2	1	1	1	1	2	2	2	2	2	2
3	1	1	1	1	3	3	3	3	3	3
4	1	2	2	2	1	1	1	2	2	2
5	1	2	2	2	2	2	2	3	3	3
6	1	2	2	2	3	3	3	1	1	1
7	1	3	3	3	1	1	1	3	3	3
8	1	3	3	3	2	2	2	1	1	1
9	1	3	3	3	3	3	3	2	2	2
10	2	1	2	3	1	2	3	1	2	3
11	2	1	2	3	2	3	1	2	3	1
12	2	1	2	3	3	1	2	3	1	2
13	2	2	3	1	1	2	3	2	3	1
14	2	2	3	1	2	3	1	3	1	2
15	2	2	3	1	3	1	2	1	2	3
16	2	3	1	2	1	2	3	3	1	2
17	2	3	1	2	2	3	1	1	2	3
18	2	3	1	2	3	1	2	2	3	1
19	3	1	3	2	1	3	2	1	3	2
20	3	1	3	2	2	1	3	2	1	3
21	3	1	3	2	3	2	1	3	2	1
22	3	2	1	3	1	3	2	2	1	3
23	3	2	1	3	2	1	3	3	2	1
24	3	2	1	3	3	2	1	1	3	2
25	3	3	2	1	1	3	2	3	2	1
26	3	3	2	1	2	1	3	1	3	2
27	3	3	2	1	3	2	1	2	1	3

are the Taguchi experiments that are tailored to their parameters and levels. The rows of the exams table represent the levels of factors in each experiment, and its columns represent the number of factors.

Using the results of experiments designed using the Taguchi method and after analyzing the results of the experiments, the optimal combination of factor levels and output voltage and cost values are calculated at the optimal point. The effect of each level is listed in

Table 4. The Average Value of Levels.

Value	Tavg (Nm)			Trip			THDU
Level	1	2	3	1	2	3	1	2	3
β1	9.80	9.82	9.79	0.43	0.41	0.41	0.41	0.37	0.39
β2	9.57	9.97	9.82	0.55	0.32	0.35	0.39	0.38	0.39
dm1	10.12	9.80	9.77	0.34	0.44	0.44	0.39	0.38	0.39
dm2	9.72	9.68	9.85	0.41	0.43	0.43	0.39	0.38	0.39
wb1	9.26	10.13	9.61	0.42	0.43	0.44	0.39	0.37	0.39
wrib1	9.64	9.99	9.73	0.38	0.41	0.39	0.35	0.42	0.36
wrib2	9.93	9.85	9.81	0.43	0.43	0.42	0.38	0.39	0.38
syoke	9.98	9.42	9.86	0.42	0.42	0.41	0.39	0.38	0.39
stooth	9.58	9.99	9.87	0.43	0.41	0.44	0.41	0.36	0.38
lso	9.82	9.88	9.83	0.41	0.45	0.43	0.38	0.38	0.38

.

Figure 11, Figure 12 and Figure 13 present the output graph for different levels of each parameter. It can be observed that T_avg is most sensitive to S_yoke and S_tooth, while T_rip is most sensitive to β₂, w_b1 and S_tooth. On the other hand, w_b1 and S_tooth have the greatest influence on THD_U.

Table 5. Comparison of the Parameters and Performances.

Parameters	Initial	Optimized by Taguchi-MEC
β2	12.0°	12.4°
wb1	3.50 mm	10.00 mm
stooth	11.00 mm	11.12 mm
syoke	11.00 mm	11.38 mm
β1	12.00°	11.10°
dm1	3.50 mm	4.41 mm
wrib1-1	0.60 mm	0.58 mm
wrib2	0.50 mm	0.28 mm
dm2	2.50 mm	3.50 mm
wrib1-2	0.60 mm	0.50 mm
lso	1.0 mm	1.0 mm

gives the initial parameters and Taguchi-MEC optimized parameters.

Table 6. Comparison of the Optimization Results Between Taguchi-MEC and FEA.

Results	Initial	Taguchi-MEC	FEA (by JMAG)
Tavg	9.40 Nm	9.82 Nm	9.90 Nm
Trip	37.4%	31.2%	30.5%
THDU	37%	35%	35%

compares the T_avg, T_rip and THD_U. As can be seen, the final results obtained by Taguchi-MEC are very close to those obtained by the multi-object optimization carried out on JMAG. It should be emphasized that the total time needed for Taguchi-MEC is 10 min (including nearly 8 min organizing data), while the time needed for JMAG is nearly 58 h (43,086 elements in mesh and 180 steps in each case). Overall, the proposed Taguchi-MEC method exhibits the great advantage of fast calculation with promising accuracy.

A brief comparison of the proposed 12/4 SynRM with existing state of the art is carried out. Figure 14 shows the 12/4 SynRM with concentrated winding (CW) and its counterpart, i.e., the 12/4 SynRM with distributed windings (DW). As can be seen, the end winding in the CW machine is much smaller than the DW one; thus, more laminations can be adopted when the same stack length is achieved. A higher copper fill factor and more coil turns are also obtained by the CW machine. More details regarding design specification and electromagnetic performances are given in

Table 7. Design Specifications of the 12/4 SynRMs.

Item	Distributed Winding	Concentrated Winding
Outer diameter	150 mm	150 mm
Lamination length	50 mm	75 mm
Total end winding	45 mm	20 mm
Total stack length	95 mm	95 mm
Turns per coil	30	40
Copper fill factor	0.5	0.58
Average torque	9.6 Nm	9.9 Nm
Torque ripple	30%	30.5%
Mutual linkage coefficient, kml	50%	14%

, where the DW SynRM is optimized by a multi-object optimization method carried out on JMAG. The fault tolerant capability is evaluated by the mutual linkage coefficient k_ml, which is also given in Table 7.

$$k_{ml}=\frac{{\sum }_{\theta =0}^{360}\left(\left|M_{ab}\right|\right)}{{\sum }_{\theta =0}^{360}\left(L_{aa}\right)}$$

(19)

where θ is electrical rotor position. The smaller k_ml indicates weaker mutual linkage between armature phases and thus better fault tolerant capability. As can be seen from Table 7, the proposed 12/4 CW SynRM shows a better torque output result and much better k_ml than the traditional 12/4 DW SynRM. The advantage of the torque performances of the CW machine is more obvious when the total stack length is smaller than 95 mm since the proportion of lamination length to total stack length varies with available space in stack.

6. Experimental Validation

The experimental setup shown in Figure 15 consists of a prototyped 12/4 SynRM and a back-to-back load machine, which are both controlled by inverters. The load machine of the experimental platform supports the torque range of 50 Nm and the max speed of 15,000 r/min. The inverter supports the power of 30 kW/60 A. The torque sensor supports the torque range of 50 Nm/10 Nm with the accuracy of 0.1%, respectively. The torque shown in Figure 16 is measured by a sensor positioned between the SynRM and the load machine. Figure 17 compares the torque values obtained by MEC and FEA, and as can be seen, good agreements are achieved. The FEA calculated losses and efficiency is given in

Table 8. Power loss and efficiency at rated operation.

Item	FEA Results
Iron loss	15.7 W
Copper loss	130.8 W
Efficiency	91.40%
Power factor	0.65
Speed	1500 r/min

, and the measured efficiency under different rotor speed is presented in Figure 18.

7. Conclusions

This paper demonstrates the electromagnetic performance of a 12/4 SynRM with concentrated windings. The non-linear MEC model of the 12/4 SynRM is built to get the air-gap flux density, coil flux-linkages, armature winding inductances, etc. The saturation effect is counted in by the MEC. The MEC model is validated by 2D-FEA. Then, the parameter optimization method based on Taguchi and MEC is proposed, aiming at the best average torque, lowest torque ripple and THD in phase voltage. Overall, the proposed Taguchi-MEC method exhibits comparable accuracy with traditional FEA-based multi-object parameter optimization methods, but using much less time. It should be emphasized that the MEC is not used to deal with complex topologies such as a rotor with a non-uniform rotor iron shape or rotor outline. Nevertheless, the proposed MEC-Taguchi method can be employed to obtain a roughly optimized topology that can be used as the initial design to be further optimized by FEA.