Table 1 shows the design condition of the basic model of the spoke-type PMSM for the washing machine. The diameter of rotor is 60% of the diameter of the stator. The coil is 0.8 mm diameter of aluminum and teeth concentrated winding was applied with a fill factor of 0.45.

Table 1
Design condition of the basic model for washing machine.

The washing machine has a gear with a ratio of 11.6:1, as shown in Table 1, connected to the spoke-type PMSM. Table 2 shows the required output condition of the spoke-type PMSM for the washing machine.

Table 2
Required output of the spoke-type PMSM.

On the washing mode, a pulsator of the washing machine rotates at 300 rpm when the spoke-type PMSM rotates at 3480 rpm. For dehydrating mode, a washing machine tub rotates at 740 rpm when the spoke-type PMSM rotates at 8584 rpm. The required torque should be 1.12 Nm for washing mode and 0.80 Nm for dehydrating mode. The back-EMF should not exceed the voltage limit of 161 V by the DC link according to Table 1. Fig. (1) shows the torque-speed curve of the constant power characteristic of the basic model.

Fig. (1) Torque and speed curve of spoke-type PMSM for a washing machine.

In this paper, the stator shape is not modified and the rotor shape is re-designed to reduce the manufacturing cost and improve the power density. Fig. (2a) presents the basic model and Fig. (2b) shows the distribution of magnetic flux density at 3480 rpm. It is well known factor that magnetic flux leakage decreasing the power and the efficiency occurred and saturated on the inner rib and outer rib in shown the Fig. (2b). Hence, this is a method of improving the torque by reducing the magnetic flux leakage, while satisfying the rotor stiffness at high-speed.

Fig. (2) Basic model of the spoke-type PMSM for a washing machine.

Fig. (3) shows the parameters of slit shape and notch shape. Parts of re-design are divided into two parts, one is slit shape and another one is notch shape. First, making the groove with the slit shape applied to the outer rib. Second, making the groove of the notch shape at both ends of the permanent magnet with thickness of 5 mm.

In case of slit shape grooves, and are the width and length of the slit shape applied to the outer rib portion, respectively. The slit shape width is in the radial direction from the end of the rotor to the center of the rotor. In addition, the slit shape length is on both sides with respect to the center of the permanent magnet. The next step is to make the groove of the notch.

Fig. (3) Design variables of the slit shape and the notch shape.

Shape, so that is the center of the notch shape and 0.5 mm away from the end of the permanent magnet. and denote inner length and outer length of the notch shape, respectively, centered on the center of notch shape. represents the notch shape width from the end of the rotor to the center of the rotor in the radial direction. indicates the inner rib, which is also the area where magnetic flux leakage and magnetic flux density saturation occur.

In order to increase torque, the inner rib of the rotor was partially removed. This is because the torque increases by removing the inner rib, which affects magnetic flux leakage [18X. Ge, Z.Q. Zhu, J. Li, and J. Chen, "A spoke-type IPM machine with novel alternate airspace barriers and reduction of unipolar leakage flux by step-staggered rotor", IEEE Trans. Ind. Appl., vol. 52, no. 6, pp. 4789-4797.
[http://dx.doi.org/10.1109/TIA.2016.2600649] ]. The re-design of the rotor flowchart is shown in Fig. (4).

Fig. (4) Flow chart of optimizing shape of slit and notch.

First, we optimize the width and length of the slit shape to improve the torque under the condition that the safe factor is 1.3 or more. Second, the width, inner length, and outer length of the notch shape were optimized when both the safe factor and the torque improvement are satisfied in the same design. In order to improve the power density with the optimized shape of slit and notch, stack length is reduced to meet the required torque and output condition of the spoke-type PMSM for a washing machine. The formula for the safe factor of the motor is expressed in Eq. (1). The ultimate stress of the electric steel sheet used for the washing machine motor is 467 MPa.

(1)

In order to optimize the slit shape applied to the outer rib portion, optimization was performed for the width of the slit shape at first. For reference, the outer rib width of the base model is 0.5 mm. As stated forehead, the width of the slit shape has the direction that enters the center of the rotor in the radial direction from the end of the rotor. Simulation test was carried out and the torque was obtained by changing the slit shape width from 0 mm to 0.35 mm. In this case, the slit shape length is fixed at 4 mm. Fig. (5a) shows that the torque at 3480 rpm increases as the slit shape width increases.

Fig. (5) Torque at 3480rpm by various width and length of slit shape.

It can be seen that a tendency that the air-gap magnetic flux density transmitted to the air gap increases as the leakage magnetic flux decreases. Fig. (5b) shows the torque according to the change of the slit shape length with the slit shape width of 0.33 mm optimized in previous step. As the slit shape length increases, the torque increases, but at 7 mm, the torque tends to decrease. The reason is that when the slit shape length is 7 mm, magnetic resistance increases that affect of the magnetic flux transmitted to the air-gap. Therefore, when the slit shape length is 6 mm, the maximum torque can be determined to be 1.29 Nm at 3480 rpm. In Fig. (6), the torque on dehydrating mode is 0.80 Nm @ 8584 rpm and the back-EMF is 161.0V, satisfying both the required output conditions.

In the stiffness analysis, maximum stress occurs around the corner of ribs when a rotor is rotating at high speed. Thickness of the ribs should be appropriate thickness that the permanent magnet is prevented from scattering. Considering the safe factor, the actual stress is 391.1 MPa is shown in Table 3 when the slit shape width is 0.34 mm. It means that the safe factor is 1.19, which cannot meet the required safe factor of 1.3. On the another hand, the slit shape width of 0.33 mm meets the required safe factor and the torque is 1.27 Nm at 3480 rpm. For the result of stiffness analysis with the optimized slit shape, the actual stress is 313.36 MPa at 8584 rpm as shown in Fig. (7) and the safe factor is 1.49.

Fig. (6) Torque at 8584 rpm with optimized slit shape.

Table 3
Result of stiffness analysis by 0.33 mm and 0.34 mm of slit shape width.

Fig. (7) Result of stiffness analysis at 8584 rpm with the optimized slit shape.

In the 3.2 session, we optimized the width and length of the slit shape and the torque improved by 15.2% compared to the base model. As a final shape, additional torque enhancement was carried out by adding a notch shape to the optimized slit shape. The inner length of notch shape was fixed at 1 mm to optimize the width of notch shape. Fig. (8a) shows the torque along the changes of the notch shape width.

Fig. (8) Torque at 3480 rpm by various width and inner length of notch shape.

A constant torque of 1.29 Nm at 3480 rpm was obtained with the notch shape width of 0.1 mm to 0.5 mm. However, when the notch shape width is 0.6 mm, the torque increases to 1.30 Nm at 3480 rpm and saturates. The reason for the increase in torque at the notch shape width of 0.6 mm is that the area of the outer rib related to the q-axis inductance decreases as shown in Table 4. When the notch shape width is 0.6 mm, the inductance difference is found to be between the d-axis and q-axis, which generates the reluctance torque, and the reluctance torque decreases according to Eq. (2). In contrast to the reluctance torque phenomenon, the magnetic torque is analyzed to be relatively larger than the reluctance torque as the air gap magnetic flux linkage increases.

Table 4
Inductance changes by notch shape width.

(2)

Where P is the number of magnetic poles, ψ_a is the linkage flux by the permanent magnets, i_d is the current of d-axis, i_q is the current of q-axis, L_d is the inductance of d-axis, and L_q is the inductance of q-axis.

Therefore, the notch shape width of 0.6 mm is the optimal value, and torque is analyzed through various outer lengths of the notch shape in the next step. In Section 3.2, when the slit shape length is 7 mm or more, the magnetic resistance is increased affecting the flow of magnetic flux transmitted to the air-gap and the torque decreased. In a similar trend, when the outer length of the notch shape is simulated from 0 mm to 0.5 mm, the change in torque is not apparent, so the notch shape outer length of 0 mm is determined. For the next step, Fig. (8b) shows the result of optimizing the inner length of the notch shape with the optimized width and outer length of the notch shape. When the inner length of the notch shape is 0.5 mm, the torque is 1.30 Nm at 3480 rpm and saturated. The torque on the dehydrating mode is 0.80 Nm at 8584 rpm and the back-EMF is satisfied as 160.7 V.

Fig. (9) shows the final model with the optimized slit and notch shape. Comparision of torque between the basic and final model is shown in Fig. (10). The torque improved 16% from 1.12 Nm to 1.30 Nm. The result of stiffness analysis at 8584 rpm, the dehydrating mode of the final model is shown in Fig. (11). As a result of the stiffness analysis of the final model, the actual stress is 330.83 MPa and the safety factor is 1.4, satisfying the requirement of 1.3.

Fig. (9) Final optimized model of spoke-type PMSM for a washing machine.

Fig. (10) Torque of final model compared with the basic model.

Fig. (11) Result of stiffness analysis of the final model.

In Sections 3.2 and 3.3, the shape of slit and notch were optimized and the torque increases to 1.30 Nm at 3480 rpm, which is increased to16% as compared to the basic model. In order to enhance the power density, it is necessary to reduce the stack length of the final model to meet the output requirements of the motor for a washing machine. Reducing the stack length can improve the power density and save manufacturing costs. The required torque of the washing machine motor is 1.12 Nm at 3480 rpm and 0.80 Nm at 8584 rpm, as shown in Table 2. Fig. (12) represents the torque analysis by varying the stack length to obtain the the required torque.

Fig. (12) Torque of Final model at 3480 rpm by various stack length.

The spoke-type PMSM generates the magnetic torque and the reluctance torque. When the stack length reduces, the torque decreases linearly as follows [19J.N. Bae, " Permanent magnet synchronous machine design through an automatic selection of the specific loadings", Ph.D. Thesis, . Hanyang University, Seoul, South Korea, 2010.];

(3)

Where K_w is the winding factor, B_g1 is the fundamental wave magnitude of air-gap magnetic flux density, ac is the electric loading, D_g is the diameter of air gap, L_stk is the stack length, G_a1 is fundamental wave magnitude of relative air-gap permeance.

The final stack length satisfying the required output of the spoke-type PMSM for a washing machine is 40 mm, which is 6 mm smaller than the basic model. The output characteristics of the basic model and the final model is shown in Table 5. The power density of the final model on the washing mode was improved by 15% and the efficiency by 0.7% compared with the basic model. The power density on the dehydration mode was improved by 14.5% and the efficiency was improved by 0.2%.

Table 5
Comparing output characteristic with the basic model and the final model.