## 2 Proposed PMSM-SCS Based on DTC Strategy
### PMSM-SCS Based on DTC Strategy Using Constant Flux
#### 2.1.2 Observer for Speed, Electromagnetic Torque, and Flux Estimations  
The sensorless nature of the PMSM-SCS based on the DTC strategy is given by the fact that an observer is used to estimate rotor speed, electromagnetic torque and flux. In this sense, a coordinate transformation between the \(d\)-\(q\) rotating reference frame and the \(\alpha\)-\(\beta\) stationary reference frame for the quantities describing the PMSM operating equations is as follows [34]:  
\[ \begin{bmatrix}f_{d}\\ f_{q}\end{bmatrix} = \begin{bmatrix}\cos\theta_{e}&\sin\theta_{e}\\ -\sin\theta_{e}&\cos\theta_{e}\end{bmatrix}\begin{bmatrix}f_{\alpha}\\ f_{\beta}\end{bmatrix}  \tag{14}\]  
where \(f_{\alpha}\) and \(f_{\beta}\) are the variables in the \(\alpha\)-\(\beta\) stationary reference frame, and \(f_{d}\) and \(f_{q}\) are the variables in the \(d\)-\(q\) rotor reference frame. In Equation (14), the coupling matrix uses the electrical angle \(\theta\), whose relation to the mechanical angle \(\theta\) representing the position of the PMSM rotor is \(\theta_{e}=n_{p}\theta\). Using relation (14), the following equations can be written in the \(\alpha\)-\(\beta\) stationary reference-frame coordinate system to describe the operation of the PMSM. \[ \begin{bmatrix}u_{\alpha}\\ u_{\beta}\end{bmatrix} = \begin{bmatrix}R+pL_{d}&\omega_{e}\big{(}L_{d}-L_{q} \big{)}\\ -\omega_{e}\big{(}L_{d}-L_{q}\big{)}&R+pL_{d}\end{bmatrix}\begin{bmatrix}i_{ \alpha}\\ i_{\beta}\end{bmatrix} +\left\{\big{(}L_{d}-L_{q}\big{)}\bigg{(}\omega_{e}i_{d }-\frac{di_{q}}{dt}\bigg{)}+\omega_{e}\lambda_{0}\right\} \begin{bmatrix}-\sin \theta_{e}\\ \cos\theta_{e}\end{bmatrix}  \tag{15}\]  
The back-EMF voltages \(e_{\alpha}\) and \(e_{\beta}\) are defined as follows [7]:  
\[ \begin{bmatrix}e_{\alpha}\\ e_{\beta}\end{bmatrix} =\bigg{\{}\big{(}L_{d}-L_{q}\big{)}\bigg{(}\omega_{e}i_{ d}-\frac{di_{q}}{dt}\bigg{)}+\omega_{e}\psi_{f}\bigg{\}} \begin{bmatrix}-\sin \theta_{e}\\ \cos\theta_{e}\end{bmatrix}  \tag{16}\]The \(i_{\alpha}\) and \(i_{\beta}\) current time variation may be derived from Equation (15), in the following form:  
\[ \begin{bmatrix}\frac{di_{\alpha}}{dt}\\ \frac{di_{\beta}}{dt}\end{bmatrix} = \begin{bmatrix}-\frac{R}{L_{d}}&-\frac{ \omega_{e}\big{(}L_{d}-L_{q}\big{)}}{L_{d}}\\ \frac{\omega_{e}\big{(}L_{d}-L_{q}\big{)}}{L_{d}}&-\frac{R}{L_{d}} \end{bmatrix}\begin{bmatrix}i_{\alpha}\\ i_{\beta}\end{bmatrix} - \begin{bmatrix}\frac{1}{L_{d}}&0\\ 0&\frac{1}{L_{d}}\end{bmatrix}\begin{bmatrix}e_{\alpha}\\ e_{\beta}\end{bmatrix} +\frac{1}{L_{d}} \begin{bmatrix}u_{\alpha}\\ u_{\beta}\end{bmatrix}  \tag{17}\]  
Similarly, the time variation of the stator flux in the \(\alpha\)-\(\beta\) stationary reference frame can be written as follows [7]:  
\[\left\{ \begin{array}{l}\frac{d\psi_{\alpha}}{dt}=u_{\alpha}-Ri_{\alpha}\\ \frac{d\psi_{\beta}}{dt}=u_{\beta}-Ri_{\beta}\end{array} \right. \tag{18}\]  
The expression for the electromagnetic torque \(T_{e}\) in the stationary reference frame \(\alpha\)-\(\beta\) is expressed as follows [7]:  
\[T_{e}=\frac{3n_{p}}{2}\big{(}\psi_{\alpha}i_{\beta}-\psi_{\beta}i_{\alpha} \big{)} \tag{19}\]  
In an obvious way, the relation between the stator flux and its projections in the \(\alpha\)-\(\beta\) stationary reference frame and in the \(d\)-\(q\) rotating reference frame is as follows:  
\[\psi^{2}=\psi_{\alpha}^{2}+\psi_{\beta}^{2}=\psi_{d}^{2}+\psi_{q}^{2} \tag{20}\]  
Starting from Equation (18), the expressions for the stator flux \(\psi= \begin{bmatrix}\psi_{\alpha}&\psi_{\beta}\end{bmatrix} ^{T}\) in the \(\alpha\)-\(\beta\) stationary reference frame are expressed as follows:  
\[\left\{ \begin{array}{l}\psi_{\alpha}=\int\limits_{0}^{t}\big{(}u_{\alpha}- R_{s}i_{\alpha}\big{)}dt\\ \psi_{\beta}=\int\limits_{0}^{t}\big{(}u_{\beta}-R_{s}i_{\beta}\big{)}dt\end{array} \right. \tag{21}\]  
\[\left\{ \begin{array}{l}\overline{\psi}_{s}=\sqrt{\tilde{\psi}_{\alpha}^{2}+ \tilde{\psi}_{\beta}^{2}}\\ \angle\hat{\psi}_{s}=arcg\frac{\hat{\psi}_{\beta}}{\hat{\psi}_{\alpha}}\end{array} \right.