## 3 Modeling the Direct-Starting Mode of an Induction Motor
The electromagnetic torque characteristics of an IM during the starting phase are illustrated for different load conditions in Figure 4. The curves correspond to varying load torques represented by the load variables SM and SMC. Specifically, SM = 0.8 (pu) represents a variable load, while SMC = 0.2 (pu) and SMC = 0.5 (pu) represent constant loads on the motor. These characteristics are influenced by different values of the motor's active and reactive resistances. The torque response demonstrates how the motor behaves under these changing load conditions, providing a detailed understanding of its dynamic performance during starting mode. Figure 2: Characteristic stages of the induction motor starting process, where Te is the electromagnetic torque and S is the slip [57]. As discussed earlier, these characteristics exhibit significant differences, reflecting the complex interactions and transient phenomena that occur during dynamic operation compared to the steady-state performance. Figure 4: Dynamic characteristics of a 3 kW induction motor during the starting mode, where Te (electromagnetic torque) and n (rotational speed) are expressed in per units, depending on the induction motor load characteristics. Figure 3: Starting mode of a 3 kW induction motor, where n (rotational frequency), I (current), and Te (electromagnetic torque) are all expressed in per unit (pu), as a function of time (in seconds). 4 Modeling the Direct Starting of an Induction Motor with Consideration of Slot Current Displacement  
In a squirrel cage induction motor, slots are the grooves in the rotor and stator cores where the conductive bars (in the rotor) and windings (in the stator) are housed. These slots are essential because they guide the flow of electrical current through the motor and contribute to generating the electromagnetic field necessary for its operation. Changes in the active resistance of the winding can be caused by variations in temperature, current displacement due to frequency changes, and the inclusion of nonlinear resistors. Current displacement in the slot refers to the uneven distribution of electric current within the conductor bars in the rotor slots. This occurs because the inductive reactance of the conductors varies depending on their position within the slot. Conductors located at the bottom of the slot experience different inductive reactance compared to those nearer the air gap. As a result, the current is not uniformly distributed along the slot height. When the angular velocity of the rotor changes, the frequency of the current in the rotor bars also changes, which further affects the distribution of current density along the conductors inside the slots. This variation in current density leads to changes in both the amplitude and phase of the currents along the slot height. In most cases, changes in active resistance due to temperature occur slowly and do not significantly affect the motor's dynamic behavior. However, when operating electric converters (ECs)--devices used to control the speed and torque of induction motors by adjusting supply frequency and voltage--the most significant focus is on studying the impact of current displacement in the slots on the motor's dynamic characteristics. The distribution of current along the slot is illustrated in Figure 6a. The current is distributed unevenly along the width of the slot as well. The coefficient \(K_{r}\), which accounts for the increase in active resistance due to current displacement, depends on the type of winding, the geometrical data of the slot, the number and sizes of the elementary conductors, and the frequency of the current in them [49]. The leakage inductive reactance also changes due to current displacement. This is accounted for by the coefficient \(K_{x}\). The coefficients \(K_{r}\) and \(K_{x}\) change according to a nonlinear law as the rotor speed varies. The changes in \(K_{r}\) and \(K_{x}\) in per unit for a deep slot are shown in Figure 6b.