# Induction Motor Dynamics Regimes: A Comprehensive Study of Mathematical Models and Validation
## 1 Introduction  
Induction motors with squirrel-cage rotors (IMs) are the most widely used type of electric motor [1]. They are relatively inexpensive and require minimal maintenance [2]. IMs are extensively utilized across various applications, including metalworking, woodworking, and general purpose machinery. They are also employed in press-forming, weaving, sewing, hoisting, and earth-moving equipment, as well as in fans, pumps, compressors, centrifuges, escalators, electric hand tools, household appliances, and numerous other devices [3, 4, 5, 6, 7]. Additionally, induction motors are commonly used in hydrogen compression systems due to their reliability, efficiency, and ease of maintenance [8, 9, 10]. It is difficult to identify any industry that does not rely on induction motors [11, 12]. According to statistical data, electric motors and motor systems in industrial and infrastructure applications with pumps, fans, and compressors in buildings are responsible for 53% of the world's total electricity consumption [13] (Figure 1). This underscores the significant role of IMs as the most widely used type of electric machine in the industrial sector. The theory of transient processes in electrical machines has been extensively developed, as demonstrated by numerous studies on this topic [14, 15, 16, 17, 18, 19]. With the increase in machine capacities and their intensified usage, there has been a growing demand for higher precision in modeling complex processes in induction motors [20, 21]. This has necessitated the refinement and development of motor calculationmethods. However, achieving both high precision and simplicity in the solutions simultaneously remains a persistent challenge [22]. Transient processes in induction motors (IM) are so diverse that it is impossible to study them comprehensively. As a result, research in this field is ongoing, as evidenced by a wide range of topics published in various sources. These studies primarily focus on the following aspects of IM operation: the starting regime [23; 24], switching mode [25; 26; 27], operation under voltage unbalance [28; 29; 30; 31], diagnostics [32; 33; 34; 35], and IM parameter identification [36; 37; 38; 39; 40; 41; 42]. Transient analysis is essential for accurately predicting dynamic behaviors such as starting currents, electromagnetic torque, and speed profiles in induction motors. This understanding enables the design of appropriate protection systems, optimized switching devices, and configurations that enhance motor and network performance. By addressing challenges such as voltage unbalance and load variations, transient analysis ensures both motor efficiency and reliable operation of power systems. Inrush currents arising in the stator and rotor windings during the starting process follow a complex oscillatory pattern [22]. The electromagnetic torque induced by these currents also exhibits oscillatory behavior and is a complex function of time [43]. Calculating the starting currents and torques of induction motors is essential for selecting appropriate protection and switching devices. Furthermore, it is crucial for assessing the permissible voltage drop in the network during motor starting, especially when the motor's capacity is comparable to that of the power supply equipment [44; 45]. Transient processes do not last long. However, their effect on electric machine operation and the overall network and drive system can be significant [18]. Transient processes are more diverse and complicated than steady-state processes, which are generally transient processes particular case. Transient processes appear in electric machines when voltage, frequency, or load changes occur, such as switching machine on or off, reversing, short circuit, or parameter changes [46; 47; 48]. These processes are influenced by various factors, and their combinations can vary widely (e.g., voltage changes, frequency, equivalent parameters, or load variations).