## 3 Simulation and Result  
The proposed method was verified through simulations conducted using MATLAB/Simulink. An induction motor was modeled using the built-in asynchronous machine module, with its rated values and parameters summarized in Table 1. The overall simulation architecture is illustrated in Figure 12.  
The simulation study compares two control strategies: the proposed field-oriented control (FOC) method and the conventional indirect FOC approach. Two separate simulation scenarios were designed. In the first scenario, only the speed command experiences a step change in \(10\pi\to 100\pi\left(\dfrac{\text{rad}}{\text{s}}\right)\) at \(t=1\) s. In the second scenario, only the torque load is subjected to a step change in \(2.04\to 18.36\left(\text{Nm}\right)\) at \(t=1\) s. It is important to note that prior to the step disturbances, the system had already reached a steady-state condition under a constant speed command \(100\pi\left(\dfrac{\text{rad}}{\text{s}}\right)\) and a constant load torque \(18.36\left(\text{Nm}\right)\). Figure 13 presents the simulation results. In all figures, the green and brown curves represent the performance of the proposed FOC method and the conventional FOC method, respectively. The left-hand side of the figure--Figure 13a--d--corresponds to the scenario in which a step change in the speed command is applied at \(t=1\) s under a fixed load torque. Specifically, Figure 13a shows the rotor speed response. Figure 13b illustrates the variation in electromagnetic torque. Figure 13c compares the rotor flux angle estimations from both methods. It can be observed that while both methods produce similar results during transient conditions, their differences become increasingly evident once the system reaches steady state. Figure 13d shows the magnitude of the stator current vector. After the step disturbance, the current amplitude under the proposed method gradually decreases, indicating improved efficiency. The right-hand side of Figure 13e-h--illustrates the scenario in which a step change in load torque is applied at \(t=1\) s while the speed command remains constant. Figure 13e shows the rotor speed response to the sudden load variation. Figure 13f presents the corresponding electromagnetic torque. Figure 13g compares the estimated rotor flux angles obtained from the two methods. Figure 13h displays the magnitude of the stator current vector. Similar trends to those observed in the speed-command variation case can be seen here: during the transient period, both methods perform comparably; however, once the system stabilizes, the proposed method exhibits superior estimation accuracy and improved current efficiency. \begin{table}
\begin{tabular}{l l} \hline \hline
**Quantity** & **Rated Value** \\ \hline Power & 5 (hp) \\ Speed & 1750 (rpm) \\ Voltage & 460 (V) \\ Frequency & 60 (Hz) \\ Rotor’s inertia & J = 0.02\(\left(\text{kg}\cdot\text{m}^{2}\right)\) \\ Damping friction & \(B=0.005752\left(\text{Ns}\middle/\text{m}\right)202\) \\ Number of poles & 4 \\ Rotor’s resistance & \(r_{r}=1.083\left(\Omega\right)\) \\ Stator’s resistance & \(r_{s}=1.115\left(\Omega\right)\) \\ Inductance of flux leakage & \(L_{ls}=L_{lr}=0.005974\left(\text{H}\right)\) \\ Mutual inductance & \(L_{m}=0.2037\left(\text{H}\right)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The ratings and parameters of this modular motor. Figure 13: The results of the simulation. Subfigures (**a–d**) depict the system response under a constant load torque, where a step change in the speed command is applied at **t = 1 s**. Subfigures (**e–h**) present the system behavior under a constant speed command, with a sudden change in the load torque introduced at **t = 1 s**. Specifically: (**a,e**) show the rotor speed response; (**b,f**) illustrate the variation in electromagnetic torque; (**c,g**) compare the rotor flux angle estimated by two different algorithms; (**d,h**) represent the magnitude of the stator current phaso.