# Inductance Estimation Based on Wavelet-GMDH for Sensorless Control of PMSM
## 2 Magnetic Saturation of Sensorless Control  
Generally, the voltage equations in the \(d\)-\(q\) synchronous frame for the vector control of PMSM is given by  
\[ \begin{bmatrix}v_{d}\\ v_{q}\end{bmatrix} = \begin{bmatrix}R_{a}+pL_{d}&-\omega L_{q}\\ \omega L_{q}&R_{a}+pL_{d}\end{bmatrix}\begin{bmatrix}i_{d}\\ i_{q}\end{bmatrix} + \begin{bmatrix}0\\ \omega\psi\end{bmatrix} , \tag{1}\]  
where \(R_{a}\), \(\psi\), and \(p\) are the stator resistance, the back EMF constant, and a differential operator, respectively.\(L_{d}\), \(L_{q}\), \(v_{d}\), \(v_{q}\), and \(i_{d}\), \(i_{q}\) are the inductances, the stator voltages, and currents in the \(d\)-\(q\) synchronous reference frame, respectively. Ignoring the difference between the estimated speed and the actual speed, Equation (1) can be converted to the \(\gamma\)-\(\delta\) axes estimated by the extended back EMF. Then, the voltage equation can be represented as [1]  
\[ \begin{bmatrix}v_{\gamma}\\ v_{\delta}\end{bmatrix} = \begin{bmatrix}R_{a}+pL_{d}&-\omega L_{q}\\ \omega L_{q}&R_{a}+pL_{d}\end{bmatrix}\begin{bmatrix}i_{\gamma}\\ i_{\delta}\end{bmatrix} + \begin{bmatrix}e_{\gamma}\\ e_{\delta}\end{bmatrix} , \tag{2}\]  
where \(e_{\gamma}=-E_{ex}sin\theta_{e},e_{\delta}=-E_{ex}cos\theta_{e}\), and \(E_{ex}=\omega\left[\left(L_{d}-L_{q}\right)i_{d}+\psi\right]-\left(L_{d}-L_{q }\right)\frac{di_{q}}{dt}\). \(v_{\gamma}\), \(v_{\delta}\), \(i_{\gamma}\), \(i_{\delta}\), and \(e_{\gamma}\), \(e_{\delta}\) represent the stator voltages, the currents, and the extended back EMFs in the \(\gamma\)-\(\delta\) estimated synchronous reference frame, respectively, accounting for the rotor position error. Utilizing \(e_{\gamma}\) and \(e_{\delta}\), the rotor position error can be calculated as  
\[\hat{\theta}_{e}=tan^{-1}\left(-\frac{\hat{\theta}_{\gamma}}{\hat{\theta}_{ \delta}}\right)\cong-\frac{\hat{e}_{\gamma}}{E_{ex}}. \tag{3}\]  
As shown in Figure 1, the estimated speed \(\hat{\omega}_{r}\) can be obtained through a PI controller with the estimated rotor position error \(\hat{\theta}_{e}\) as the input and a low-pass filter for noise reduction. Moreover, the estimated rotor position \(\hat{\theta}\) can be obtained by integrating the estimated speed. Figure 1: Block diagram of extended EMF based sensorless control algorithm. Ensuring precise sensorless control requires accurately determining parameters such as the resistance and inductance, as represented in Equation (2). In systems affected by significant periodic loads, like those in pumps or compressors, the discrepancy between the actual \(q\)-axis inductance (\(L_{q}\)) and the estimated \(q\)-axis inductance (\(\hat{L}_{q}\)) due to magnetic saturation leads to inaccuracies in the estimated rotor position. Consequently, to address the deviations resulting from magnetic saturation between the actual and estimated values, Equation (2) can be rearranged for the extended back EMF \(e_{\gamma}\), \(e_{\delta}\) as  
\[ \begin{bmatrix}e_{\gamma}\\ e_{\delta}\end{bmatrix} = \begin{bmatrix}v_{\gamma}\\ v_{\delta}\end{bmatrix} - \begin{bmatrix}R_{\text{ }}+pL_{d}&-\omega\hat{L}_{q}\\ \omega\hat{L}_{q}&R_{\text{ }}+pL_{d}\end{bmatrix}\begin{bmatrix}i_{\gamma}\\ i_{\delta}\end{bmatrix} + \begin{bmatrix}\Delta e_{\gamma}\\ \Delta e_{\delta}\end{bmatrix} , \tag{4}\]  
where \(\Delta e_{\gamma}=-p\Delta L_{d}i_{\gamma}+\omega\Delta L_{q}i_{\delta}\), and \(\Delta e_{\delta}=-p\Delta L_{d}i_{\delta}-\omega\Delta L_{q}i_{\gamma}\). Note that \(\Delta L_{d}\) and \(\Delta L_{q}\) are the errors of the \(d\)- and \(q\)-axes inductance, which are the value obtained by subtracting the estimated inductance from the actual inductance. \(e_{\gamma}\) and \(e_{\delta}\) increase or decrease due to the influence of \(\Delta e_{\gamma}\) and \(\Delta e_{\delta}\) including the inductance error, in a practical system. As a result, the rotor position error and speed ripple increase.