## 4 Verification by Simulations  
In order to verify the correctness and effectiveness of the proposed method, the simulation models were built according to the IPMSM sensorless control system based on the traditional high-frequency voltage injection method and the proposed rotating high-frequency voltage injection method in the MATLAB 2014a software. It is assumed that both the motor and the operating environment are ideal. (1) The motor core is not saturated. This assumption means that there is a linear relationship between the magnetic field and winding currents. (2) The motor has a completely symmetrical magnetic circuit and winding. This assumption includes the following aspects: the three-phase windings of the stator are identical, the space positions are separated by \(2/3\pi\) arc, and the excitation windings of each pole of the rotor are identical. (3) The self-inductance magnetic field of the three-phase stator winding and the mutual inductance magnetic field between the stator and the rotor winding are distributed in a sinusoidal manner along the air gap. The back EMF of the motor is a sine wave. In the simulation models, the estimated rotor position and speed obtained by the position sensorless control system are used to replace the measured rotor position and speed, respectively. To compare the detection accuracy and dynamic performance of the sensorless algorithm, the simulation results of the two methods are shown in the following figures. Figure 7 shows the experimental results of the traditional method from forward 50 r/min to reverse 50 r/min. Figure 7a is the given speed, which changes from forward 50 r/min to reverse 50 r/min. Figure 7b,c show the actual position \(\theta_{e}\) and estimated  
Figure 6: Amplitude-frequency and phase-frequency characteristic curves of the SFF. position \(\theta_{e}\), respectively. In addition, Figure 7d presents the rotor position error. From the figure, it can be seen that the steady-state error is about 0.002 rad at 50 r/min. In the process of the speed change, the rotor position error becomes larger, and the maximum rotor position error is 0.055 rad at the moment of speed change. Figure 8 shows the experimental results of the proposed method from forward 50 r/min to reverse 50 r/min. Figure 8a-c show the actual position \(\theta_{e}\), estimated position \(\dot{\theta}_{e}\) and the rotor position error, respectively. The experimental results show that the  
Figure 7: Experimental results of traditional method from forward 50 r/min to reverse 50 r/min: (a) Rotor speed; (b) Actual rotor position; (c) Estimated rotor position; (d) Rotor position error. steady-state error is about 0.001 rad at 50 r/min, and the maximum rotor position error is 0.04 rad at the moment of speed change. The rotor position error comparison curves of the traditional method and the proposed method during the speed change are shown in Figure 9. It can be seen that the error of the proposed method is smaller than that of the traditional method. Especially when the speed changed, the error of the proposed method is 0.15 rad smaller than that of the traditional method. The error comparison curves of the two methods during the load change are shown in Figure 10. Figure 10a is the given load, and Figure 10b is the rotor position error comparison curves. It can be seen that the error of the proposed method is about 0.02 rad smaller than that of the traditional method when the load changed. In addition, to analyze the system's robustness, Figure 11 shows the error comparison curves of different resistance values. Take the winding resistance value R1 as 0.566 ohms and R2 as twice that of R1. The simulation results show that the errors of different resistance values are almost equal, no matter whether in a steady state or a dynamic state. The same results could be obtained when the motor pole pairs or inductance parameters changed. Therefore, the changes in parameters do not affect the steady-state and transient performance of the system.