## 3 Modeling the Direct-Starting Mode of an Induction Motor  
The direct-starting mode of an IM is a critical stage in its operation, as it determines whether the motor can successfully transition from rest to its steady-state operating condition [37]. The method of starting plays a crucial role in ensuring that the motor operates effectively, and different starting methods are employed depending on the operating conditions and the electrical network requirements [50]. When using direct starting, however, large inrush currents and significant voltage drops in the network can occur, which may reduce the motor's lifespan and affect the stability of the electrical system. During the starting process, the application of voltage to the stator winding induces both harmonic currents and flux linkages corresponding to the applied voltage, as well as transient components. These transients result in complex interactions between the stator and rotor fluxes. With some simplification, it can be assumed that one of the transient fluxes remains stationary relative to the stator, while the other rotates with the rotor. The forced flux rotates at synchronous frequency, and the rotor flux linkage caused by the rotating-free flux decays over time [51; 52]. The electromagnetic torque generated during this phase is the result of the interaction of all these fluxes. The transient currents that arise in the stator and rotor windings during the starting process follow complex oscillatory patterns, and the electromagnetic torque they generate is similarly oscillatory and dependent on time [53; 54]. Understanding the processes that occur during starting allows for the selection of optimal starting methods to minimize these negative effects [7]. There are two major challenges associated with the direct starting mode of an induction motor (IM): a high starting current, which can be 6-10 times higher than the normal operating current, and fluctuations in the motor's starting torque. These torque fluctuations, or variations, mean that the motor does not produce smooth torque during startup, which can result in vibrations and mechanical stress [6; 55; 56]. These factors lead to the following consequences:  
* The high starting current causes a significant voltage drop on the main substation buses (especially if the transformer and motor power are comparable), which disrupt the operation of other consumers and the motor itself (resulting in a start-up delay). * The high starting current also induces significant thermal overload on the windings, leading to accelerated aging of the insulation. This ultimately results in insulation breakdown and inter-turn short circuits. * Large torque fluctuations during the start-up period, which can exceed the nominal torque by 4-5 times, create unfavorable conditions for the mechanical system connected to the motor (such as gears, belts, or shafts), causing stress and potentially damaging these components. The process of accelerating an IM during starting mode is divided into three stages (Figure 2):  
1. Application of voltage to the stator windings, while the rotor is in a fixed position, inducing both forced and transient currents in the windings. 2. The rotor begins to rotate and reaches critical slip, which is the point where the motor produces its maximum torque. This occurs when the rotor speed lags behind the synchronous speed of the magnetic field by an optimal amount before the motor accelerates further. 3. The rotor accelerates from critical slip to a steady rotational speed. For starting-mode modeling, the parameters of 3 kW induction motor with the following per unit (pu) values were used [58]: \(X_{1}\)= 0.057 pu, \(X_{ad}\) = 3.4 pu, \(X_{2}\) = 0.1 pu, \(R_{1}\)= 0.072 pu, \(R_{2}\) = 0.0487 pu, \(T_{M}\) = 32.986. Figure 3 presents the dynamic characteristics of torque, current, and rotation frequency obtained through mathematical modeling for the direct-starting mode of an IM under no-load operation.