## 2 New Field-Oriented Control (FOC) Technique  
Therefore, the objective of this paper is to accurately estimate the rotor flux angle (\(\theta_{rf}^{s*}\)) under steady-state conditions using only directly measurable stator current and voltage signals, while avoiding integration operations and adopting a continuously correctable approach. Thismethod addresses the limitations of conventional FOC schemes, particularly the estimation inaccuracies that become more pronounced at low speeds. The estimated rotor flux angle is directly incorporated into the FOC algorithm to enhance real-time performance. According to the extended formulation presented by [PERSON] [3], the control equation of the stator circuit for a squirrel cage induction motor, expressed in the rotating d-q reference frame with angular velocity \(\omega\), is given by  
\[ \begin{split} v^{\omega}_{ds}&=r_{s}\ddot{u}^{ \omega}_{ds}+D\lambda^{\omega}_{ds}-\omega\lambda^{\omega}_{qs}\;,\lambda^{ \omega}_{ds}=L_{s}\dot{u}^{\omega}_{ds}+L_{m}\ddot{u}^{\omega}_{dr}\\ v^{\omega}_{qs}&=r_{s}\ddot{u}^{\omega}_{qs}+D \lambda^{\omega}_{qs}+\omega\lambda^{\omega}_{ds}\;,\lambda^{\omega}_{qs}=L_{s} \dot{u}^{\omega}_{qs}+L_{m}\ddot{u}^{\omega}_{qr}\end{split}  \tag{6}\]  
where \(r_{s}\) and \(L_{s}\) denote the stator resistance and inductance, respectively. The corresponding rotor circuit equation is expressed in  
\[ \begin{split} v^{\omega}_{dr}&=r_{r}\dot{u}^{ \omega}_{dr}+D\lambda^{\omega}_{dr}-(\omega-\omega_{r})\lambda^{\omega}_{qr}= 0\;,\lambda^{\omega}_{dr}=L_{r}\dot{u}^{\omega}_{dr}+L_{m}\ddot{u}^{\omega}_ {ds}\\ v^{\omega}_{qr}&=r_{r}\dot{u}^{\omega}_{qr}+D \lambda^{\omega}_{qr}+(\omega-\omega_{r})\lambda^{\omega}_{dr}=0\;,\lambda^{ \omega}_{qr}=L_{r}\dot{u}^{\omega}_{qr}+L_{m}\ddot{u}^{\omega}_{qs},\end{split}  \tag{7}\]  
where \(D\equiv\dfrac{d}{dt}\) represents the time derivative operator. Substituting the magnetic flux expressions from (6) and (7) into the voltage equations yields  
\[ \begin{split} v^{\omega}_{qs}&=\omega L_{s}\ddot{u }^{\omega}_{ds}+(r_{s}+L_{s}D)\ddot{u}^{\omega}_{qs}+\omega L_{m}\ddot{u}^{ \omega}_{dr}+L_{m}D\ddot{u}^{\omega}_{qr}\\ v^{\omega}_{ds}&=(r_{s}+L_{s}D)\ddot{u}^{\omega} _{ds}-\omega L_{s}\ddot{u}^{\omega}_{qs}+L_{m}D\ddot{u}^{\omega}_{qr}-\omega L _{m}\ddot{u}^{\omega}_{qr}\\ v^{\omega}_{dr}&=L_{m}D\ddot{u}^{\omega}_{ds}-( \omega-\omega_{r})L_{m}\ddot{u}^{\omega}_{qs}+(r_{r}+L_{r}D)\dot{u}^{\omega}_{ dr}-(\omega-\omega_{r})L_{r}\dot{u}^{\omega}_{qr}=0\\ v^{\omega}_{qr}&=(\omega-\omega_{r})L_{m}\ddot{u}^{ \omega}_{ds}+L_{m}D\ddot{u}^{\omega}_{qs}+(\omega-\omega_{r})L_{r}\dot{u}^{ \omega}_{dr}+(r_{r}+L_{r}D)\ddot{u}^{\omega}_{qr}=0\end{split} . \tag{8}\]  
\(\lambda^{rf}_{qr}\equiv 0\Rightarrow D\lambda^{rf}_{qr}=0\) is obtained under the assumption that the d-axis of the rotating reference frame is continuously aligned with the rotor flux vector. Under this alignment condition, the original stator voltage Equation (6) simplifies to the following set of reduced equations:  
\[0=r_{r}\dot{r}^{rf}_{dr}+D\lambda^{rf}_{dr}, \tag{9}\]  
\[0=r_{r}\dot{r}^{rf}_{qr}+\Big{(}\omega_{rf}-\omega_{r}\Big{)}\lambda^{rf}_{ dr}, \tag{10}\]  
\[\lambda^{rf}_{dr}=L_{r}\dot{r}^{rf}_{dr}+L_{m}\dot{u}^{rf}_{ds}, \tag{11}\]  
\[0=L_{r}\dot{r}^{rf}_{qr}+L_{m}\dot{r}^{rf}_{qs}. \tag{12}\]  
From Equation (11), an intermediate variable \(\dot{r}^{rf}_{dr}=\dfrac{\lambda^{rf}_{dr}-L_{m}\dot{r}^{rf}_{ds}}{L_{r}}\) is derived and subsequently substituted into Equation (9), yielding the following:  
\[(1+\tau_{r}D)\lambda^{rf}_{dr}=L_{m}\dot{r}^{rf}_{ds}\Rightarrow\lambda^{rf}_ {dr}=\dfrac{L_{m}}{1+\tau_{r}s}\dot{r}^{rf}_{ds}. \tag{13}\]  
In (13), \(\tau_{r}\triangleq\
icefrac{{L_{r}}}{{r_{r}}}\) denotes the rotor time constant, and \(s\) represents the Laplace operator. It is important to note that (13) holds only under relaxed motor conditions (i.e., steady-state operation).