*2.1. Design Target and Parameters*

For the SBW application, a three-phase 12/8 SRM topology is chosen. A greater pole number is generally associated with higher torque, and a lower number corresponds to a higher rotational speed [22]. In SBW, the motor is not running continuously, but only on command according to the driver. The automotive voltage of 12 V is used as the input supply. The maximum current should be lower than 30 A to reduce the burden of the converter. Detailed specifications of the proposed motor are shown in Table 1. Another important aspect of the application is bi-directional rotation: the SRM must be able to rotate clockwise and counterclockwise to shift the gear back and forth. Most non-uniform air gap SRMs cut a small part of the rotor/stator tip in the same way for every pole. However, this method disables the bi-directional rotating capability since the inductance profile becomes asymmetrical. A solution to this is presented below.



### *2.2. Conventional 12*/*8 SRM*

Common SRMs have a uniform air-gap, as shown in Figure 2a. The notations β*s*, β*r*, and β*<sup>g</sup>* denote the stator pole arc, rotor pole arc, and the angular distance between the stator and rotor pole tips when fully unaligned. Zero position is as determined in the figure as the position at which the rotor is completely unaligned with the stator. The corresponding torque and inductance profiles can be seen in Figure 2b. The inductance profile depends on the geometry of the motor and determines the torque as follows:

$$T\_{\rm pl} = \frac{1}{2} t\_{\rm pl}^2 \frac{dL\_{\rm pl}}{d\Theta} \tag{1}$$

where *Tph*, *iph*, *Lph*, and θ are the phase torque, current, inductance, and rotor position. *Lph* can be obtained by one-phase excitation, and this value changes according to rotor position. Considering the linear magnetic characteristic, in SRM, the torque is proportional to the square of current and the change of inductance. The straight line is at 22.5◦, where both poles are perfectly aligned and both left and right sides are identical to each other. This symmetry ensures bi-rotational rotation capability. The average torque considering the positive torque area only is 0.324 Nm.

**Figure 2.** Conventional 12/8 SRM: (**a**) 2D model and (**b**) characteristics.

The goal of this research is to reduce the torque ripple and shape the waveform close to an ideal profile, as shown in Figure 1b, whereas square-shaped torque is generated with constant current excitation. In Figure 2b, there is a protruding section between P and Q where the torque appears to have some "overshoot" before flattening again at Q. The angular distance between P and Q is 6◦.
