## 3 Modeling the Direct-Starting Mode of an Induction Motor
By determining the laws of variation for the active and reactive resistances of the slot, the equations of electromechanical energy conversion are modeled, taking into account the changes in the rotor-winding resistances. Figure 5: Static and dynamic starting characteristics of a 3 kW induction motor. Research shows that the nonlinear variation of the rotor's active resistance has the greatest impact on the dynamics during the starting mode of an IM. Due to current displacement in the rotor slots, the acceleration time is reduced, and surge currents and torques are decreased. The shape of the grooves affects the law of change in \(K_{r}\) and \(K_{x}\) and, consequently, impacts the dynamic characteristics. However, the initial and final values of resistance have a greater influence on the energy conversion processes. To determine the parameters of resistance, change during the startup of an induction motor, the following expression can be used [59]:  
\[Z_{2}^{\prime}(s)=\frac{R_{2}^{\prime}(s)}{s}+jX_{2}^{\prime}(s) \tag{25}\]  
where \(R_{2}^{\prime},\ X_{2}^{\prime}(s)\) are rotor parameters that depend on slip, reduced to the stator frequency, i.e., \(X_{2}^{\prime}(s)=\omega_{1}L_{2}^{\prime}(s)\). In all design types of squirrel cage rotors (deep-bar, double-cage, and solid), the active resistance increases with increasing slip, while the reactive resistance decreases. It can be shown that the same approximate formula for parameters as a function of slip is valid for all types of rotors [59]. \[R_{2}^{\prime}(s)=R_{2N}^{\prime}\bigg{[}1+\bigg{(}\frac{R_{2}^{\prime}(1)}{R _{2N}^{\prime}}-1\bigg{)}\sqrt{s}\bigg{]} \tag{26}\]  
\[X_{2}^{\prime}(s)=X_{2a}^{\prime}\Bigg{[}1+\frac{\bigg{(}\frac{X_{2N}^{\prime }}{\frac{X_{2a}^{\prime}}{\frac{1}{X_{2}^{\prime}}}}-1\bigg{)}}{1+b\sqrt{s}} \Bigg{]} \tag{27}\]  
\[b=\frac{\frac{X_{2N}^{\prime}-1}{\frac{X_{2}^{\prime}(1)}{X_{2}^{\prime}(1)}} }{1-\frac{X_{2}^{\prime}}{X_{2}^{\prime}(1)}} \tag{28}\]  
where \(R_{2N}^{\prime},\ X_{2}^{\prime}(1)\) rotor windings' parameters at slow frequency (practically on nominal slip): \(R_{2}^{\prime}(1),\ RX_{2}^{\prime}(1)\) parameter on slip \(s=1\). \(X_{2a}^{\prime}\)--the part of reactive rotor winding, which is not depending on slip (differential dispersion or dispersion of crown of tooth, dispersion in the open slot, dispersion of end coil, except the machines with current dispersion in end coil). Using the IM mathematical model, the starting mode was modeled, taking into account current displacement in slots. The modeling results are presented on Figures 7-9. The curves also reflect the characteristics obtained during starting mode without considering current displacement in the slots, as discussed in the previous chapter. Figure 6: Current distribution along the height of the slot in an electrical machine (**a**); dependence of \(K_{r}\) and \(K_{x}\) on slip (**b**). Figure 2:0 Rotation frequency change characteristics on IM starting mode. \(n\)—starting without current displacement in slots. \(n^{\prime}\)—starting mode taking current displacement in the slots. Figure 2:1 Current change characteristics during IM starting mode. I—starting mode without considering current displacement in slots; I—starting mode with current displacement in slots. Figure 2:2 Electromagnetic torque change characteristics during IM starting. \(T_{e}\)—starting without considering current displacement in slots; \(T_{e}{}^{\prime}\)—starting with current displacement in slots taken into account. scope was used for data acquisition, with the data transferred to a data management system for further analysis. The setup is powered by a three-phase AC source, with transformers (TR and ATR) providing necessary adjustments for the measurements. The induction motor's T-type equivalent circuit is shown in Figure 12[49]. Table 1 lists the data of the tested IM, while Table 2 presents the data of the IM equivalent circuit (Figure 11) used for modeling [58].