## 2 Optimized Geometric and Operational Design of an Induction Motor
### Finite Element Method (FEM) Simulation
As for saturation, the red area near the stator and in the slots may indicate areas reaching magnetic saturation levels. In this case, control was required as saturation can reduce efficiency and increase losses. Based on an analysis of the methodology, optimization, sensitivity analysis, and final geometric characteristics of the SCIM for high-power (10 MW) ship propulsion, it is essential to discuss the distinctive aspects of designing and manufacturing a Wound-Rotor Induction Motor (WRIM) for similar applications. The WRIM is distinguished by its rotor construction, where externally accessible slip rings and brushes enable variable rotor resistance control. This design feature offers distinct advantages over SCIMs in demanding environments. The ability to adjust the rotor resistance dynamically allows for better control of slip, making WRIMs particularly suitable for variable-load applications such as ship propulsion. Compared to SCIMs, WRIMs can provide a higher starting torque with lower inrush current, thereby reducing mechanical and electrical stress on ship power systems. Copper losses in the rotor are typically higher in WRIMs due to external resistance insertion. However, this can be mitigated by optimizing the resistance value to balance efficiency and performance. The general equation for the induced torque in a WRIM is derived from the equivalent circuit (Figure 1) and is given by Equation (13). \[\mathrm{T}_{\mathrm{ind}}=\frac{3V_{\mathrm{th}}^{2}}{\omega_{\mathrm{s}}s} \cdot\frac{\mathrm{R}_{r\mathrm{\ total}}}{\left(\mathrm{R}_{\mathrm{th}}+ \mathrm{R}_{\mathrm{rotal}}/s\right)^{2}+\left(\mathrm{X}_{\mathrm{th}}+ \mathrm{X}_{r}\right)^{2}} \tag{13}\]  
where \(\mathrm{V}_{\mathrm{th}}=\mathrm{V}_{\mathrm{s}}\frac{\mathrm{X}_{M}}{\sqrt{ \mathrm{R}_{\mathrm{s}}^{2}+\left(\mathrm{X}_{\mathrm{s}}+\mathrm{X}_{M} \right)^{2}}}\), \(\mathrm{R}_{\mathrm{th}}+\mathrm{j}\mathrm{X}_{\mathrm{th}}=\frac{\mathrm{j} \mathrm{X}_{M}\left(\mathrm{R}_{\mathrm{s}}+\mathrm{j}\mathrm{X}_{\mathrm{s} }\right)}{\mathrm{R}_{\mathrm{s}}+\mathrm{j}\mathrm{X}_{\mathrm{M}}}\), \(\mathrm{R}_{\mathrm{rotal}}=\mathrm{R}_{r}+\mathrm{R}_{\mathrm{ext}}\), \(\mathrm{R}_{\mathrm{s}}\) is the stator resistance, \(\mathrm{R}_{r}\) the rotor resistance, \(\mathrm{X}_{\mathrm{s}}\) the stator reactance, \(\mathrm{X}_{\mathrm{r}}\) the rotor reactance, \(\mathrm{X}_{\mathrm{M}}\) the magnetizing reactance, and \(\mathrm{R}_{\mathrm{ext}}\) the external resistance. To ensure optimal torque performance in a WRIM, the external rotor resistance must be carefully adjusted. While an increase in \(\mathrm{R}_{\mathrm{ext}}\) can improve starting torque by increasing rotor slip and reducing inrush currents, excessive values can lead to a reduction in both starting torque and steady-state performance. According to the Maximum Power Transfer Theorem, the maximum torque at startup is achieved when the total rotor resistance is equal to the rotor reactance at the slip corresponding to peak torque. For starting conditions (i.e., \(\mathrm{s}=1\)), the external resistance should be chosen such that \(\mathrm{R}_{\mathrm{ext}}=\mathrm{X}_{\mathrm{r}}-\mathrm{R}_{r}\). This ensures that the motor develops maximum starting torque without excessive energy dissipation in the rotor circuit. If \(\mathrm{R}_{\mathrm{ext}}\) is too high, the denominator in the torque equation increases disproportionately, leading to a decrease in torque. Conversely, if \(\mathrm{R}_{\mathrm{ext}}\) is too low, the starting torque remains suboptimal due to an imbalance between resistance and reactance. Thus, the ideal tuning of \(\mathrm{R}_{\mathrm{ext}}\) depends on the motor's design parameters, particularly the rotor resistance and reactance. In practical applications, a stepped resistance control or external resistor bank is often used to dynamically adjust \(\mathrm{R}_{\mathrm{ext}}\) during startup and operation, ensuring that the motor achieves high torque during acceleration while minimizing power losses under steady-state conditions.