# An NUTSF at Sub-Region for Suppression of Torque Ripple in Switched Reluctance Motors
## 4 Proposed NUTSF Control Strategy
### New Region Allocation and Proposed TSF  
The TSF allocates the torque reference to each phase winding according to the value of the waveform in the TpE, thus the torque reference of each phase winding is obtained. The waveform of the TSF directly affects the performance of torque ripple suppression in the TSF control strategy. A suitable TSF can not only reduce torque ripple, but also reduce current amplitude, which will improve the efficiency of the SRM. Therefore, the TpE is further divided into two regions according to the change rate of inductance, and the new TSF will be given for the incoming phase and the outgoing phase in each region.  
A region allocated diagram is shown in Figure 6 based on the inductance characteristics, where \(\theta_{on}\) is the turn-on angle, \(\theta_{m}\) is the boundary point of the large inductance, \(\theta_{on}+\theta_{ov}\) is the sum of the turn-on angle and the overlapping angle, and \(\theta_{off}\) is the turn-off angle. Taking phase B as an example in Figure 6, the inductance of the incoming phase rises in the TpE, but the inductance change rate \(\partial L(k)/\partial\theta\) is small in the first half of the TpE and increases intensely in the second half. Therefore, the TpE is divided into two regions, namely, region 1 and region 2, and the point where the inductance obviously rises is defined as the inductance boundary point \(\theta_{m}\). Region 1 is between \(\theta_{on}\) and \(\theta_{m}\), and region 2 is between \(\theta_{m}\) and the \(\theta_{on}+\theta_{ov}\). The TpEs of the other two phases are allocated according to the same rule.  
According to the inductance characteristics in region 1 and 2, a non-unity torque sharing function (NUTSF) is designed based on each sub-region, as shown in Formula (7), in order to improve the conventional exponential TSF. Taking \(\theta_{m}\) as the boundary point, each TpE is subdivided into two regions in the proposed TSF where function \(q_{up}(\theta)\) and function \(q_{down}(\theta)\) in Formula (7) are located, and the TSFs \(q_{up1}(\theta)\), \(q_{up2}(\theta)\), \(q_{down1}(\theta)\) and \(q_{down2}(\theta)\) are given in each region.  
\(q_{up1}(\theta)\) and \(q_{down1}(\theta)\) in region 1 produce the torque of the incoming phase and outgoing phase, respectively, and \(q_{up2}(\theta)\) and \(q_{down2}(\theta)\) in region 2 produce the torque of the incoming phase and outgoing phase, respectively. The NUTSF can be expressed as follows  
\[\left\{\begin{array}{ll}q_{1}(\theta)&=0&0\leq\theta\leq\theta_{on}\\ q_{up1}(\theta)&=\left[e^{\frac{k}{-(\theta-\theta_{on})}+1}\right]^{P1}& \theta_{on}\leq\theta\leq\theta_{m}\\ q_{up2}(\theta)&=\left[e^{-\frac{\theta_{ov}}{-(\theta-\theta_{on})}+1} \right]^{P2}&\theta_{m}\leq\theta\leq\theta_{on}+\theta_{ov}\\ q_{2}(\theta)&=1&\theta_{on}+\theta_{ov}\leq\theta\leq\theta_{off}\\ q_{down1}(\theta)&=1-\left[e^{-\frac{k}{-(\theta-\theta_{on})}+1}\right]^{P1} &\theta_{off}\leq\theta\leq\theta_{m}\\ q_{down2}(\theta)&=1-\left[e^{-\frac{\theta_{on}}{-(\theta-\theta_{on})}+1} \right]^{P2}&\theta_{m}\leq\theta\leq\theta_{off}+\theta_{ov}\\ q_{3}(\theta)&=0&\theta_{off}\leq\theta\leq\tau\end{array}\right. \tag{7}\]  
where \(\theta_{m}\) is the boundary points, \(\tau\) is the rotor pole pitch, \(k\) is the exponential constant, \(\theta_{ov}\) is the overlapping angle, \(\theta_{on}\) is the turn-on angle, \(\theta_{off}\) is the turn-off angle, \(P1\) is the \(P1\) th-power of the function in region 1 and \(P2\) is the \(P2\) th-power of the function in region 2.  
Comparing Formula (6) with (7), a TSF is divided into four when the NUTSF control strategy is used according to the torque characteristics of region 1 and 2. It can clearly be seen that the TSFs in each region are not united. The non-unity function will bring different function shapes, and the further adjustment of the function shapes depends on the th-power of \(P1\) and \(P2\).