## 1 Introduction
Also, the study of the operating modes of an IM plays a crucial role in optimizing its operation to reduce energy consumption [32]. Optimal management of operating modes and a deep understanding of the processes occurring within the motor contribute to increased energy efficiency. This is particularly important in situations where automatic control systems are absent. Knowledge of the electrical and thermal processes within the motor helps prevent emergency situations and ensures safe operating conditions [33]. Research and optimization of operating modes without the use of automatic control systems can significantly reduce operating and maintenance costs. Proper operation of the motor reduces expenses on maintenance and replacement of worn components, positively affecting the economic efficiency of equipment use [34]. The study of dynamic operating modes of the induction motor, taking into account electromagnetic transient phenomena caused by changes in the motor's state, is a highly relevant task in the modern practice of using low-inertia induction drives. Low-inertia induction drives refer to systems with reduced rotational inertia, which allows for faster acceleration and more responsive operation to changes in load or speed. These drives are particularly useful in applications where quick dynamic responses are required, such as robotics or precision machinery [35]. Direct starting, which involves applying full voltage to the motor, is the simplest and most cost-effective method of starting an IM [36]. It is also the most commonly used method for starting equipment, particularly in applications without automatic control systems, due to its simplicity and cost-effectiveness [37]. A deeper understanding of dynamic processes, such as electromagnetic transient phenomena, not only enhances the efficiency of the motor's operation but also improves its reliability. This, in turn, reduces the risk of accidents and extends the lifespan of the equipment, which is crucial for enterprises aiming for sustainable development and reduced operational costs. Thus, a comprehensive approach to studying the operating modes of the induction motor provides significant advantages, contributing to the overall efficiency and reliability of production processes. To model IM-starting behavior, particularly in direct-start conditions, several mathematical models are commonly employed. These models range in complexity and accuracy,depending on the level of detail they capture and the assumptions they make. The most commonly used mathematical models for simulating the start-up behavior of induction motors provide varying levels of insight, depending on the phenomena they are designed to represent [38; 39]. One of the simplest methods for analyzing induction motor startup behavior is the steady-state equivalent circuit model, which represents the motor's stator and rotor characteristics using electrical resistances and inductances. While computationally efficient, it assumes steady-state conditions and ideal sinusoidal voltages, limiting its ability to capture dynamic processes during startup [40; 41; 42]. A more advanced approach is the dynamic D-Q model, which uses Park's transformation to simplify analysis by converting three-phase currents into two orthogonal components in a rotating reference frame. This model is effective for capturing transient phenomena, like inrush currents and torque oscillations, though it requires precise rotor parameters and balanced conditions [12; 39; 43; 44]. Coupled magnetic circuit models represent the motor as multiple coupled circuits and are especially useful for simulating rotor bar currents during startup but require significant computational resources [45]. Space vector models simplify the analysis of induction motors by using complex vectors to represent the motor's electrical variables. This method provides a compact and efficient way to model motor dynamics during both steady-state and transient conditions.