# Research on Influence of Switching Angle on the Vibration of Switched Reluctance Motor
## 2 Principle of Switching Angle control Method
\tag{1}\]  
Since the voltage drop of the motor winding is very small, the influence of the voltage drop of the motor winding is neglected, and the circuit equations at both ends of the winding are simplified to:  
\[U_{k}=\frac{d\psi_{k}}{dt}=\frac{d\psi_{k}}{d\theta}\cdot\frac{d\theta}{dt}= \frac{d\psi_{k}}{d\theta}\cdot\omega \tag{2}\]  
where \(U_{k}\) is the voltage at both ends of the winding, and \(\psi_{k}\) is the flux linkage at both ends of the winding, and the transformation Formula (2) is:  
\[d\psi_{k}=\frac{U_{k}}{\omega}\cdot d\theta \tag{3}\]  
It can be seen from Formula (3) that in the linear model of the motor, when the motor runs at a constant speed, the flux linkage of the winding changes linearly with the slope of \(U_{k}/\omega\) along with the rotor angle. When the phase winding of the switched reluctance  
Figure 1: (**a**) The simplified structure of a 12-8 SRM (**b**) Inductance curve of linear model. motor is turned on, the corresponding rotor angle is the turn-on angle \(\theta_{on}\), the rotor angle when it is turned off is the turn-off angle \(\theta_{off}\), and the conduction angle is \(\theta_{c}=\theta_{off}-\theta_{on}\). When the flux linkage of the winding conduction is 0 and the source voltage is \(U_{k}\), then the expression of each phase flux linkage relative to rotor angle can be calculated by integrating both sides of Formula (3) as:  
\[\psi_{k}=\int_{\theta_{on}}^{\theta}\frac{U_{k}}{\omega}\cdot d\theta=\frac{U_ {k}}{\omega}\cdot(\theta-\theta_{on}) \tag{4}\]  
When \(\theta=\theta_{off}\), the maximum value of the flux linkage is:  
\[\psi_{\max}=\frac{U_{k}}{\omega}\cdot(\theta_{off}-\theta_{on})=\frac{U_{k}}{ \omega}\cdot\theta_{c} \tag{5}\]  
When the power is turned off, the winding will freewheel the power supply through the freewheeling diode, and the voltage flowing through the winding is \(U=-U_{s}\). Substitute the voltage into the Equation (3) and integrate it to obtain the freewheeling phase winding flux linkage expression as:  
\[\psi_{k}=-\int_{\theta_{on}}^{\theta}\frac{U_{k}}{\omega}\cdot d\theta=\frac{ U_{k}}{\omega}\cdot(2\theta_{off}-\theta-\theta_{on}) \tag{6}\]  
When \(\theta=2\theta_{off}-\theta_{on}\), the flux linkage decreases to zero. Through the above analysis, the variation curve of the flux linkage with the rotor angle can be obtained, as shown in Figure 2:  
Then the functional expression of flux linkage is:  
\[\psi_{k}=\left\{ \begin{array}{ll}\frac{U_{k}}{\omega}\cdot(\theta-\theta_{on }),&\theta_{on}\leq\theta<\theta_{off}\\ \frac{U_{k}}{\omega}\cdot(2\theta_{off}-\theta-\theta_{on}),&\theta_{off}\leq \theta<2\theta_{off}-\theta_{on}\\ 0,&\text{other}\end{array} \right. \tag{7}\]  
From the relationship between flux linkage and inductance, we can get:  
\[\psi=L(\theta)\cdot i(\theta) \tag{8}\]  
Substituting Equation (8) into Equation (2):  
\[U_{k}=\frac{d\psi_{k}}{dt}=L\cdot\frac{di}{dt}+i\cdot\frac{dL}{dt}=L\cdot\frac {di}{d\theta}\cdot\omega+i\cdot\frac{dL}{d\theta}\cdot\omega \tag{9}\]  
Figure 2: Flux linkage curve of linear model. Multiplying current on both sides of Equation (9) yields the following equation:  
\[U_{k}\cdot i=L\cdot i\frac{di}{d\theta}\cdot\omega+i^{2}\cdot\frac{dL}{d\theta} \cdot\omega=\frac{d}{dt}\left(\frac{1}{2}L\cdot i^{2}\right)+i^{2}\cdot\frac{dL} {d\theta}\cdot\omega \tag{10}\]  
From Formula (10), it can be seen that part of the electrical energy input of the switched reluctance motor is converted into electromagnetic field energy storage, and the other part is converted into mechanical energy, which is the multiplication of phase current and electromotive force to provide power for the motor to rotate. Figure 3 shows the variation curve of the current with the rotor position in the next electrical cycle of the angle control strategy. In the interval \(\theta_{1}\leq\theta<\theta_{on}\), the motor phase has not been turned on, and no current flows through the phase winding.