*2.3. Non-Uniform Rotor Stucture*

In order to retain the bi-directional rotating capability while maintaining a non-uniform air-gap, a mirrored structure is presented in Figure 3a. The M-type design has the cutting part on the left side, and the N-type design has this on the right side. The distribution is even for the rotor poles, meaning that there will be four "Ms" and four "Ns" for a 12/8 SRM. A similar structure was shown in [21], but instead of cutting/chipping a part of the tip, a castle-shaped pole was introduced. Compared to [21], the proposed rotor will be much simpler. The rotor surface can be divided into two sections: non-uniform and uniform. As can be seen in Figure 3b, the relationship can be expressed by the equation below, where β*non* and β*uni* are the arcs of non-uniform and uniform sections, respectively. β*non* is set to 6◦ based on previous observation.

$$
\beta\_{\overline{r}} = \beta\_{\text{nom}} + \beta\_{\text{uni}} \tag{2}
$$

**Figure 3.** Bi-directional non-uniform structure: (**a**) pole distribution and (**b**) modeling.

The cutting length, *lnon*, is measured from the pole tip vertically along the side. Therefore, the maximum non-uniform air-gap length is the addition of conventional air-gap *lg* and *lnon*.

Inductance is at maximum when the poles are aligned. Considering counterclockwise rotation, the air-gap increases gradually on the M-type pole. Generally, there are two ways to cut the pole: line and arc. The straight (blue) and concave (red) lines in Figure 3b show the difference between these approaches. Nevertheless, both cuts cause a slower increase of inductance compared to the

conventional method. Because of this, the torque is also reduced during β*non*, and instead of having an "overshoot", it appears much rounder. The β*non* section ends at point Q, and the profile after that is similar using either conventional or non-uniform methods. After alignment, the rotor pole leaves the stator pole, and due to the "N" pole, the air-gap slightly increases, thus creating symmetry in the profiles. This phenomenon is shown and compared in Figure 4. Compared to the straight non-uniform gap, the arched cut preserves some convexity in torque, which results in a greater positive area while still reducing the "overshoot" and creates a more similar feature to the ideal profile. As a result, the average torque is higher than that of the straight cut. The values of the average torque with the same current excitation are 0.324, 0.315, and 0.319 for conventional, straight, and arched cuts, respectively. Because the torque drops by 2.8% in a straight cut and 1.5% in an arched cut, the latter is chosen in this study.

**Figure 4.** Comparison of conventional and non-uniform air-gap models: (**a**) torque and (**b**) inductance.

### *2.4. Proposed Rotor Stucture*

After choosing the appropriate non-uniform section, the next step is to flatten the curve to further reduce the ripple. Figure 5 shows the design of the proposed rotor. β*disR*, β*disH*, and *ldisH* are the angular distance of the first hole to the nearest pole edge, the angular distance between the holes, and the distance from the pole surface. Note that the *ldisH* is constant regardless of the non-uniform part. Two small holes are placed near the surface because the closer they are to the shaft, the less their effect will be on torque development. Manufacturing accuracy is the only restriction in the model.

**Figure 5.** Proposed rotor structure: (**a**) pole distribution and (**b**) modeling.

In order to investigate this more deeply, the non-uniform, one-hole model is investigated. The comparison of the torque and inductance of the four designs is shown in Figure 6. Two points, R and S, mark the convex area that has to be straightened from the previous non-uniform-only structure to achieve the ideal torque waveform shape. *H*<sup>1</sup> is located at "R". It can be seen that the simple non-uniform and single hole models do not have distinct differences, as seen in the figure. However,

the corner at "R" is slightly lowered in the single-hole design. The second hole and the *H*<sup>2</sup> placement create the rather square shape, which is discussed in Section 3. In any case, the symmetrical feature is retained, and so bi-directional rotation is possible. In this study, both holes have the same diameter, but it is possible to shape them in any way to create the desired waveform in accordance with the general guidelines given here. The average torque of the one-hole and proposed model for the same current excitation is 0.319 and 0.317 Nm, respectively. In other words, the proposed method has the minimum torque compared to others, with a drop of 2.16% compared the conventional approach.

**Figure 6.** Comparison of the four models: (**a**) inductance and (**b**) torque.
