## 2 Rotating High-Frequency Voltage Injection Method
\begin{split} i_{\alpha\beta h}&=\begin{bmatrix}i_{ah}\\ i_{\beta h}\end{bmatrix}=\begin{bmatrix}I_{ph}\cos(\omega_{h}t)+I_{nh}\cos(- \omega_{h}t+2\theta_{e})\\ I_{ph}\sin(\omega_{h}t)+I_{nh}\sin(-\omega_{h}t+2\theta_{e})\end{bmatrix}\\ &=I_{ph}\mathrm{e}^{|(\omega_{h}t)}+I_{nh}\mathrm{e}^{|(-\omega_{h}t+2 \theta_{e})}\end{split}  \tag{5}\]  
where \(I_{ph}\) and \(I_{nh}\) are amplitudes of the high-frequency current positive and negative sequence components, respectively. It can be seen from the above equations that the rotor position information is contained in the current negative sequence component. To extract the rotor position information, the signal is processed according to the flow shown in Figure 2.  
where PLL is a phase-locked loop, and N/S is the identification unit of magnetic poles N and S.  
It can be seen from Figure 2 that the three-phase current of the motor is measured first. Then the three-phase current \(i_{abc}\) is transformed to the current component \(i_{a\beta}\) in the \(a\beta\) coordinate system. The high-frequency current response component \(i_{a\beta h}\) can be obtained by a BPF. The synchronous coordinate transformation is used to transform \(i_{a\beta h}\) into the coordinate system with a synchronous rotating frequency of \(\omega_{h}\). The positive sequence component of the high-frequency current becomes the direct current (DC) component in this synchronous rotating coordinate system, while the negative sequence component becomes the high-frequency component rotating in \(-2\omega_{h}\). \[i_{dqh}=\mathrm{e}^{\mathrm{i}(-\omega_{h}t)}i_{a\beta h}=I_{ph}+I_{nh} \mathrm{e}^{\mathrm{i}2(-\omega_{h}t+\theta_{c})} \tag{6}\]  
A synchronous rotating high-pass filter can be used to filter out the DC component and extract the negative sequence component \(i_{dqh}\) containing rotor position information. \[i_{dqh}=HPF\Big{[}i_{dqh}\Big{]}=HPF\Big{[}I_{ph}+I_{nh}\mathrm{e}^{\mathrm{ i}2(-\omega_{h}t+\theta_{c})}\Big{]}=I_{nh}\mathrm{e}^{\mathrm{i}2(-\omega_{h}t+ \theta_{c})} \tag{7}\]  
Using the synchronous coordinate transformation again, \(i_{dqh}\) is transformed to a coordinate system rotating synchronously in \(-2\omega_{h}\). The high-frequency negative sequence component containing the rotor position information then becomes a synchronously rotating DC component. \[i_{dqh1}=\mathrm{e}^{\mathrm{i}(2\omega_{h}t)}i_{dqh}=I_{nh}\mathrm{e}^{ \mathrm{i}(2\omega_{h})} \tag{8}\]  
As a feedback control system for adjusting phase error, the phase-locked loop includes three parts: a phase discriminator, a voltage-controlled oscillator and a loop filter. The PLL adjusts the output phase of the system by comparing the phase error between the input signal and the output signal of the voltage-controlled oscillator so that the output phase can accurately track the input phase. For the high-frequency injection-based PMSM sensorless control system, the equivalent phase-locked loop structure for position and speed estimation is shown in Figure 3. Here, \(i_{dqh1}\) is used to extract the estimated rotor position and speed information. Figure 3: Block diagram of the phase-locked loop. Figure 2: Block diagram of the signal demodulation process. The comparison unit between the actual and estimated rotor position is equivalent to the phase discriminator in the PLL. The proportional integration unit is similar to the loop filter. The primary integration unit is equivalent to the voltage-controlled oscillator part in the phase-locked loop system. The phase discriminator module of the position and speed estimation phase-locked loop uses \(i_{dh1}\), \(i_{gh1}\) and the estimated rotor position \(\hat{\theta}_{e}\) to construct a trigonometric function. The rotor position error is obtained as follows:  
\[