## 2 Rotating High-Frequency Voltage Injection Method  
The structure of the IPMSM sensorless control system based on a rotating high-frequency voltage injection method is shown in Figure 1[21; 22]. It can be seen from the figure that the rotor position estimation system includes three parts: the high-frequency signal injection module, the signal extraction part and the position observer. The high-frequency voltage is injected into the \(\alpha\beta\) coordinate system by the high-frequency injection module. A high-frequency response current is generated by the motor winding. The current contains rotor position information. The  
Figure 1: IPMSM sensorless control based on rotating high-frequency voltage injection. used to extract the high-frequency response current component and separate the signal containing the rotor position information. Then, the position observer is used to obtain the estimated rotor position and speed. The voltage equation of IPMSM in the \(\alpha\beta\) coordinate system is [28,29]  
\[ \begin{split}\left[\begin{array}{c}u_{\alpha}\\ u_{\beta}\end{array}\right]=R_{\mathrm{s}}\left[\begin{array}{c}i_{\alpha}\\ i_{\beta}\end{array}\right]+\left[\begin{array}{cc}L_{0}+L_{1}\cos(2 \theta_{e})&L_{1}\sin(2\theta_{e})\\ L_{1}\sin(2\theta_{e})&L_{0}-L_{1}\cos(2\theta_{e})\end{array}\right]\cdot p \left[\begin{array}{c}i_{\alpha}\\ i_{\beta}\end{array}\right]+\\ \omega_{e}\left[\begin{array}{cc}-L_{1}\sin(2\theta_{e})&-L_{0}+L_{1}\cos(2 \theta_{e})\\ L_{0}+L_{1}\cos(2\theta_{e})&L_{1}\sin(2\theta_{e})\end{array}\right]\cdot \left[\begin{array}{c}i_{\alpha}\\ i_{\beta}\end{array}\right]+\omega_{e}\psi_{l}\left[\begin{array}{c}-\sin \theta_{e}\\ \cos\theta_{e}\end{array}\right]\end{split}  \tag{1}\]  
where \(u_{\alpha}\) and \(u_{\beta}\) are the voltage components, \(i_{\alpha}\) and \(i_{\beta}\) are the current components in the \(\alpha\beta\) coordinate system, \(L_{0}=(L_{d}+L_{q})/2\), \(L_{1}=(L_{d}-L_{q})/2\), \(L_{\mathrm{d}}\) and \(L_{\mathrm{q}}\) are the inductance in the \(dq\) coordinate system, \(p\) is the differential operator, \(R_{\mathrm{s}}\) is the winding resistance and \(\theta_{e}\) is the rotor position. If the injected voltage frequency is much higher than the fundamental frequency, the resistance and the electric angular velocity in Equation (1) can be ignored. Therefore, the IPMSM equation can be approximated as a purely inductive model in the coordinate system as follows:  
\[ \begin{bmatrix}u_{ah}\\ u_{\beta h}\end{bmatrix} = \begin{bmatrix}L_{0}+L_{1}\cos(2\theta_{e})&L_{1} \sin(2\theta_{e})\\ L_{1}\sin(2\theta_{e})&L_{0}-L_{1}\cos(2\theta_{e})\end{bmatrix} p \begin{bmatrix} i_{ah}\\ i_{\beta h}\end{bmatrix}  \tag{2}\]  
Equation (2) can be transformed into the current differential form as  
\[ \begin{split} p\begin{bmatrix}i_{ah}\\ i_{\beta h}\end{bmatrix}&=\begin{bmatrix}L_{0}+L_{1}\cos(2\theta_{e})&L_{1} \sin(2\theta_{e})\\ L_{1}\sin(2\theta_{e})&L_{0}-L_{1}\cos(2\theta_{e})\end{bmatrix}^{-1}\begin{bmatrix} u_{ah}\\ u_{\beta h}\end{bmatrix}\\ &=\frac{1}{L_{0}^{2}-L_{1}^{2}}\begin{bmatrix}L_{0}-L_{1}\cos(2\theta_{e})&-L_{1 }\sin(2\theta_{e})\\ -L_{1}\sin(2\theta_{e})&L_{0}+L_{1}\cos(2\theta_{e})\end{bmatrix}\begin{bmatrix} u_{ah}\\ u_{\beta h}\end{bmatrix}\end{split}  \tag{3}\]  
The high-frequency voltage signal is expressed as  
\[ \begin{bmatrix}u_{ah}\\ u_{\beta h}\end{bmatrix} = \begin{bmatrix}-V_{h}\sin(\omega_{h}t)\\ V_{h}\cos(\omega_{h}t)\end{bmatrix} =V_{h}e^{j(\omega_{h}t+\frac{\alpha}{2})} \tag{4}\]  
where \(V_{h}\) is the amplitude of the high-frequency voltage and \(\omega_{h}\) is the electrical angular frequency of the injected high-frequency voltage. The high-frequency current component containing the rotor position information in the \(\alpha\beta\) coordinate system can be expressed as  
\[