## 2 New Field-Oriented Control (FOC) Technique
Additionally, Equation (12) implies the following relationship:  
\[\dot{r}^{rf}_{qr}=-\dfrac{L_{m}}{L_{r}}\dot{r}^{rf}_{qs}. \tag{14}\]By combining (10), (13), and (14), the following expression is derived: \(S\omega_{e}(s)\)  
\[\triangleq\omega_{rf}(s)-\omega_{r}(s)=\frac{1+\tau_{r}s\frac{t_{qs}^{rf}(s)}{ \tau_{r}}}{\frac{r_{ds}^{rf}}{t_{ds}^{rf}(s)}}\Rightarrow\frac{i_{qs}^{rf}(s)}{ i_{ds}^{rf}(s)}=\frac{\tau_{r}}{1+\tau_{r}s}S\omega_{e}(s)\,.\] Substituting the final value theorem into the above yields the following:  
\[\lim_{t\to\infty}\frac{i_{qs}^{rf}(t)}{i_{ds}^{rf}(t)}=\frac{\lim_{t\to \infty}i_{qs}^{rf}(t)}{\lim_{t\to\infty}i_{qs}^{rf}(t)}=\frac{\lim_{s\to 0}i_{qs}^{rf}(s)}{ \lim_{s\to 0}i_{ds}^{rf}(s)}=\lim_{s\to 0}\frac{i_{qs}^{rf}(s)}{ \tau_{ds}^{rf}(s)}=\lim_{s\to 0}\frac{\tau_{r}}{1+\tau_{r}s}S\omega_{e}(s)=\tau_{r} \lim_{s\to 0}\omega_{e}(s)\equiv\tau_{r}\cdot y. \tag{15}\]  
Based on the geometric relationship illustrated in Figure 7, the following equation is obtained:  
\[\lim_{t\to\infty}\frac{i_{qs}^{rf}(t)}{i_{ds}^{rf}(t)}=\lim_{t\to\infty}\frac {i_{qds}(t)\sin\gamma(t)}{i_{qds}(t)\cos\gamma(t)}=\lim_{t\to\infty}\tan \gamma(t)=\tan\left[\lim_{t\to\infty}\gamma(t)\right]\Rightarrow\lim_{t\to \infty}\gamma(t)=\tan^{-1}(\tau_{r}\cdot y). \tag{16}\]  
Here, \(\gamma\triangleq\theta_{e}-\theta_{rf}\) represents the angle by which the stator current vector \(i_{qds}\) leads the rotor flux vector \(\underline{\lambda}_{ddr}\). If the d-axis of the reference frame is aligned with the stator voltage vector (\(\underline{v}_{qds}(t)\)), the variable \(v_{qs}^{p}\equiv 0\) can be obtained, and Equation (8) simplifies into the following two expressions:  
\[v_{ds}^{p}=(r_{s}+L_{s}D)i_{ds}^{p}-\omega_{e}L_{s}i_{qs}^{p}+L_{m} Di_{dr}^{p}-\omega_{e}L_{m}i_{qr}^{p} \tag{17}\] \[0=r_{r}\dot{r}^{rf}_{qr}+\Big{(}\omega_{rf}-\omega_{r}\Big{)}\lambda^{rf}_{ dr}, \tag{10}\]0 \[0=r_{r}\dot{r}^{rf}_{qr}+\Big{(}\omega_{rf}-\omega_{r}\Big{)}\lambda^{rf}_{ dr}, \tag{10}\]1 (18) \[0=r_{r}\dot{r}^{rf}_{qr}+\Big{(}\omega_{rf}-\omega_{r}\Big{)}\lambda^{rf}_{ dr}, \tag{10}\]2  
Applying the Laplace transform to both equations, the solution to (17) yields the intermediate terms \(i_{dr}^{p}(s)=\frac{sv_{ds}^{p}-\big{(}\omega_{e}^{2}L_{s}+r_{s}+L_{s}s^{2} \big{)}\beta_{e}^{p}-\omega_{e}r_{s}i_{qr}^{p}}{L_{m}\big{(}\omega_{e}^{2}+s^{ 2}\big{)}}\) and \(i_{qr}^{p}(s)=\frac{\omega_{e}r_{ds}^{p}-\big{(}\omega_{e}^{2}L_{s}+r_{s}+L_{ s}s^{2}\big{)}\beta_{e}^{p}-\omega_{e}r_{s}^{p}}{L_{m}\big{(}\omega_{e}^{2}+s^{ 2}\big{)}}\). These results are then substituted into (18) to derive the following:  
\[0=r_{r}\dot{r}^{rf}_{qr}+\Big{(}\omega_{rf}-\omega_{r}\Big{)}\lambda^{rf}_{ dr}, \tag{10}\]3  
where \(a_{1}=\frac{1}{L_{m}\big{(}\omega_{e}^{2}+s^{2}\big{)}}\big{[}(\omega_{e}- \omega_{r})L_{r}\omega_{e}+(r_{r}+L_{s}s)\big{]},a_{2}=\frac{1}{L_{m}\big{(} \omega_{e}^{2}+s^{2}\big{)}}\big{[}(r_{r}+L_{s}s)\big{(}\omega_{e}^{2}L_{s}+r_ {s}s+L_{s}s^{2}\big{)}\)  
\(+\omega_{e}(\omega_{e}-\omega_{r})r_{s}L_{r}-L_{m}^{2}s\big{(}\omega_{e}^{2}+s ^{2}\big{)}\big{]},\;\;a_{3}=\frac{1}{L_{m}\big{(}\omega_{e}^{2}+s^{2}\big{)} }\big{[}(\omega_{e}-\omega_{r})L_{r}\big{(}\omega_{e}^{2}L_{s}+r_{s}s+L_{s}s^{ 2}\big{)}-\)  
\(\omega_{e}r_{s}(r_{r}+L_{s}s)-(\omega_{e}-\omega_{r})L_{m}^{2}(\omega_{e}^{2}+s ^{2})\big{]},\;\;and\;b_{1}=\frac{1}{L_{m}\big{(}\omega_{e}^{2}+s^{2}\big{)} }\big{[}(\omega_{e}-\omega_{r})L_{r}s-(r_{r}+L_{s})\omega_{e}\big{]},\;\;b_{2 }=-a_{3},\,b_{3}=-a_{2}\) are system-dependent coefficients. \[0=r_{r}\dot{r}^{rf}_{qr}+\Big{(}\omega_{rf}-\omega_{r}\Big{)}\lambda^{rf}_{ dr}, \tag{10}\]4 The geometric meaning of \(\rho\).