# Induction Motor Dynamics Regimes: A Comprehensive Study of Mathematical Models and Validation
## 4 Induction Motor Starting Mode Modeling  
To calculate the starting mode of an induction motor (IM), including current, rotational frequency, and electromagnetic torque under no-load conditions, the following IM models were used: model I--expressed in the \(\alpha\),\(\beta\) coordinate system, model II--in the \(d\_q\) coordinate systems using currents, model III--in the \(d\_q\) coordinate systems using flux linkages and model IV--in the \(d\_q\) coordinate system using flux linkages with the rotor winding electromagnetic time constant. For starting mode modeling, the parameters of a 3 kW induction motor with the following per unit (pu) values were used [76]: \(X_{1}\) = 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 5 illustrates the rotational frequency characteristics of a 3 kW induction motor during starting mode under no-load conditions. The graph shows the rotor's acceleration profile, demonstrating how the rotational frequency stabilizes as the motor reaches a steady-state operation. Figure 6 illustrates the current drawn by the induction motor during direct starting under no-load conditions. The graph highlight the initial high inrush current, which gradually decreases as the motor transitions to steady-state operation. Figure 7 shows the variation in electromagnetic torque during the starting phase of the induction motor. The torque curve emphasizes the transient behavior before stabilizing at its no-load steady-state value. Figure 5: Rotational frequency during the starting mode of a 3 kW induction motor under no-load conditions. Figure 6: Current drawn during direct starting of a 3 kW induction motor under no-load conditions. Mathematical modeling of the IM starting mode, based on the models presented earlier in this paper, reveals that regardless of the coordinate system chosen (\(\alpha\),\(\beta\) or \(d\),\(\mu\)) or the mathematical basis (currents, flux linkages, or flux linkages with rotor windings' electromagnetic time constant), the modeling results are consistent and accurately represent real-world behavior. The results of the starting mode mathematical modeling using Model I were compared with experimental data from a test setup described in [22], which included measurements of current, rotational speed, and torque using specialized equipment. These comparisons demonstrated close alignment between modeled and experimental results, validating the model's accuracy and reliability for analyzing IM starting dynamics. The data tables in Appendix A show that the values of current, rotational frequency, and electromagnetic torque are in close agreement across the four models, particularly at later time instances. However, minor discrepancies are observed during the initial period, which can be attributed to transient effects. Despite these variations, the models converge to similar steady-state behavior, indicating that any of these models can be effectively used to simulate the dynamic regimes of induction motors, provided that the transient period is appropriately considered. In addition to the no-load starting characteristics presented in Figures 5-7, the four presented IM models can also simulate starting regimes for induction motors of various capacities, with different load characteristics and torque profiles. This flexibility ensures broad applicability of these models for analyzing and optimizing motor performance under diverse operational conditions. A comparative summary of the four mathematical models is presented in Table 1. The table outlines the coordinate systems, key parameters simulated, and the strengths and limitations of each model. This summary provides additional context for understanding the unique features and applications of the models, helping to identify their suitability for different dynamic regimes of induction motor operation. Figure 7: Electromagnetic torque variation during the starting phase of a 3 kW induction motor under no-load conditions.