# FE-Aided Synchronization Analysis of Line-Start Synchronous Reluctance Motors
## 3 Analysis of the Synchronization Process  
LS-SynRMs are subjected to oscillatory phenomena near the synchronous speed during the start of synchronization. This results in a crucial problem with this motor, a problem often called synchronization. The synchronization process of a LS-SynRM can be realized in a way similar to that of a LSPM. A simplified process for LSPMs using the approximate phenomenon of torque angle dependency during the last pole slip is given in [15]. The last pole slip of a LS-SynRM is depicted in Figure 2a. From the critical slip, which is the turning point of the rotor velocity, the motor eventually reaches Point A, where the motor achieves synchronous speed for the first time. The load angle at that moment is an unstable load angle, and the net torque is zero at that moment. After passing the unstable equilibrium, the torque pulsates until it finally stabilizes at point R, as shown in Figure 2b. The trend of a LS-SynRM during synchronization can be considered identical with that of a LSPM. The torque expression in (8), which divides the torque components by the average and the oscillating torque, is fitted to the synchronization process model as follows. The momentary torque can be expressed by:  
\[T_{e}(s,\delta)=T_{0}(s)+T_{1}(s)\sin(2\delta) \tag{10}\]  
For simplification, the torque phase angle, \(\alpha\), is set to \(-\pi/2\). The critical slip \(s_{\text{cr}}\), and the unstable load angle \(\delta_{s}^{\prime}\) are defined by:  
\[T_{e}(s_{\text{cr}},\delta_{\text{sc}}) =T_{L}(s_{\text{cr}}), \tag{11}\] \[T_{e}(0,\delta_{s}^{\prime}) =T_{L}(0),\]  
where \(T_{L}\) is the load torque and \(\delta_{\text{sc}}\) is the load angle at the critical slip. In a successful pull-in process of a LSPM, the span of the load angle is approximated as \(\delta_{\text{sc}}=\delta_{s}^{\prime}-\pi\)[15]. Following the same line as that of a LSPM, the pull-in of a LS-SynRM may occur during one-half of \(\pi\) electric radians in the load angle because the angular speed of the oscillating torque is \(2s\omega\). It can be rearranged to:  
\[\delta_{\text{sc}}=\delta_{s}^{\prime}-\frac{\pi}{2} \tag{12}\]  
Therefore, the trend of the slip during the synchronization process of the LS-SynRM may be expressed by the following:  
\[s=s_{\text{cr}}\sin(\delta_{s}^{\prime}-\delta). \tag{13}\]  
The pull-in energy provided by a motor (which is the energy needed to pull the rotor from the critical slip to synchronous speed) is [15]:\[K_{p}=\int_{\xi_{\alpha_{\rm cr}}}^{\delta^{y}_{\rm s_{\alpha}}}(T_{0}(s)+T_{1}(s, \delta)-T_{L}(s))\ d\delta. \tag{14}\]  
Substituting for the slip using (13) yields the torque expression as a function of the load angle. For successful synchronization, the following limit condition must be satisfied:  
\[K_{p}\geq K_{\rm scr}, \tag{15}\]  
where \(K_{\rm s_{\alpha}}\) is the apparent rotor kinetic energy at the critical slip. \[K_{\rm s_{\alpha}}=\frac{1}{2p}J(s_{\rm cr}\omega)^{2}, \tag{16}\]  
where \(J\) is the inertia. The critical inertia is realized when the apparent rotor kinetic energy and the pull-in energy are equivalent. The estimation of the critical inertia followed by the process described above has been compared with that of the dynamic analysis results by solving the equivalent circuit portrayed in Figure 3. The arbitrary chosen parameters of a LS-SynRM referred by a IE4 3.7 kW, 4-poles IM for the analysis are given in Table 1. The parameters are obtained by a commercial software SPEED. The motor is supplied by 220 Vrms and 60 Hz sinusoidal voltage. The analysis results are shown in \[T_{e}(s_{\text{cr}},\delta_{\text{sc}}) =T_{L}(s_{\text{cr}}), \tag{11}\]0 As can be seen, discrepancies of the estimated critical inertia between the two analysis results are minor. \[T_{e}(s_{\text{cr}},\delta_{\text{sc}}) =T_{L}(s_{\text{cr}}), \tag{11}\]1 Trend of slip and alternating torque versa load angle during the synchronization process: (**a**) sine-like trend during the last pole slip (from point B to A) and (**b**) approximate corresponding alternating torque.