# An Improved Hybrid Control Scheme of a Switched Reluctance Motor for Torque Ripple Reduction
## 2 The Modeling of SRM and TSF Control
### Torque Sharing Function  
The most prominent feature of SRM is the commutation operation. During commutation, the switch of the former phase (\(p\)) is off, and the output torque decreases. The latter phase (\(p+1\)) is conductive, and the output torque increases. The goal of the TSF design is to make the increased torque exactly offset the decreased torque; thus, the total output torque can remain constant. A TSF control strategy combining hysteresis control is employed to allocate the given torque to each phase based on the distribution function. Then, the synthetic instantaneous torque can track the given torque output by the speed controller [13; 14; 15; 16; 19]. The control block of the TSF strategy is shown in Figure 2.  
Figure 1: Magnetization curves.  
The curve shape and expression of distribution function determine the inhibition effect of torque ripple and the waveform of phase current. The four conventional TSFs are linear, exponential, cubic, and cosine. The sum of each phase TSFs at any time must be constant at 1, so the following equation is satisfied.  
\[\left\{\begin{array}{l}\sum\limits_{p=1}^{N_{p}}f_{p}(\theta)=1,0\leq f_{p}( \theta)\leq 1\\ {T_{p}}^{*}={T^{*}}f_{p}(\theta)\end{array}\right. \tag{6}\]  
where \({T_{p}}^{*}\) and \(f_{p}\) are the reference phase torque and distribution function of phase \(p\), respectively.  
However, owing to the slow current response, the torque distribution function cannot be tracked well during commutation, resulting in a relatively large torque ripple. The inductance of the latter phase (\(p+1\)) has a low rate of change, resulting in low phase torque. In addition, the former phase (\(p\)) cannot be rapidly attenuated to the given torque, owing to the presence of continuing current. The TSF cannot be tracked well in actual operation, causing a high-torque ripple. Therefore, a modified piecewise TSF is proposed to reduce the torque ripple in the commutation interval. Figure 3 shows the typical waveform of cosine and the proposed TSFs, where \(\theta_{on}\), \(\theta_{off}\), and \(\theta_{ov}\) are the turn-on angle, turn-off angle, and overlap angle, respectively. The reference torque is increased to compensate for the low phase torque of the latter phase (\(p\)+1) compared with cosine TSF. The proposed TSF reduces the given torque to accelerate the decay of the former phase (\(p\)). The proposed TSF is expressed as  
\[{f_{p}}^{*}(\theta)=\left\{\begin{array}{ll}0,&0\leq\theta\leq\theta_{\text{ on}}\\ (\theta-\theta_{\text{on}})/\theta_{\text{ov}},&\theta_{\text{on}}\leq\theta \leq\theta_{\text{on}}+0.5\theta_{\text{ov}}\\ 1-2(\theta_{\text{on}}+\theta_{\text{ov}}-\theta)^{2}/{\theta_{\text{ov}}}^{2},&\theta_{\text{on}}+0.5\theta_{\text{ov}}\leq\theta\leq\theta_{\text{on}}+ \theta_{\text{ov}}\\ 1,&\theta_{\text{on}}+\theta_{\text{ov}}\leq\theta\leq\theta_{\text{off}}\\ 1-(\theta-\theta_{\text{off}})/\theta_{\text{ov}},&\theta_{\text{off}}\leq \theta\leq\theta_{\text{off}}+0.5\theta_{\text{ov}}\\ 2(\theta_{\text{off}}+\theta_{\text{ov}}-\theta)^{2}/{\theta_{\text{ov}}}^{2},&\theta_{\text{off}}+0.5\theta_{\text{ov}}\leq\theta\leq\theta_{\text{off}} +\theta_{\text{ov}}\\ 0,&\theta_{\text{off}}+\theta_{\text{ov}}\leq\theta\leq\tau\end{array}\right. \tag{7}\]  
where \(\tau\) is the rotor pole pitch.  
Figure 2: The TSF control block strategy.