## 1 Introduction  
An induction motor (IM) is an electrical machine designed to convert electrical energy into mechanical energy [1; 2]. An AC induction motor consists of a fixed outer stator and a rotor that rotates inside, separated by an air gap. In a three-phase AC induction motor, the AC supply creates a naturally rotating magnetic field in the stator. This changing magnetic field induces an electromagnetic force (EMF) in the rotor, generating currents and magnetic fields within it. The interaction between the stator and rotor magnetic fields generates torque, causing the rotor to rotate [3]. The most widely used type of IM is the squirrel cage rotor motor. Its widespread use is due to several advantages over other types of motors: high reliability, the ability to operate directly from an AC power supply, ease of maintenance, simplicity in manufacturing, low cost, and the absence of mechanical contact with the stationary parts of the machine, which ensures durability and reduces maintenance costs [4]. Three-phase induction motors with squirrel cage rotors have several advantages over single-phase induction motors, particularly in terms of performance and application. Although both single-phase and three phase induction motors use squirrel cage rotors, three-phase motors are generally more efficient, can start heavier loads, and are better suited for industrial applications. Three-phase motors also provide smoother operation and greater reliability in high-power environments, making them more suitable for large-scale commercial and industrial use compared to their single-phase counterparts [3]. The main problem encountered when using an IM with a squirrel cage rotor is the direct starting of the motor. A significant drawback of the squirrel cage rotor is its low starting torque and high intrusion current, which is several times higher than the nominal current value. These factors can negatively impact the motor's operation and cause significant mechanical shocks [5; 6; 7]. The primary applications of induction motors (IMs) without automatic control include industrial installations where cost-effective and reliable solutions are necessary. In certain settings, such as small and medium enterprises, the decision not to use automatic control systems is often driven by cost-effectiveness and practicality rather than the size of the business itself [8]. Simple and robust IMs are commonly used to drive equipment such as pumps, fans, and conveyors in these situations [9]. This also applies to agricultural settings, where induction motors power machines without the need for automation, as the simplicity and reliability of these systems make them ideal [10]. In household and utility services, such as water supply compressors and ventilation systems [11], IMs often operate without complex control systems, as the demand for automation is minimal in these applications. Understanding the performance of IMs in such scenarios is critical to ensuring their reliability and efficiency in environments where automatic control is not utilized. Research into the operating modes of IM is of significant importance, as evidenced by numerous ongoing studies in this area [12]. Understanding these modes is not only crucial for addressing starting characteristics but also for a wide range of issues, including fault detection and diagnosis. Current research focuses on fault detection in IMs [13; 14; 15], condition monitoring and practical diagnostics [16; 17], and fault diagnosis based on current signature analysis [18; 19]. Additionally, studies explore condition monitoring and diagnostics of rotor faults [20; 21; 22; 23], broken bar detection [24; 25; 26; 27; 28], and fault detection based on the start-up current envelope [29]. These investigations complement the study of IM-starting characteristics by improving the overall efficiency of IM drives through flux and torque control, which remains a key area of research [30; 31].