## 2 Optimized Geometric and Operational Design of an Induction Motor
### Preliminary Design Stage  
The goal in the first step was to establish the motor's basic geometrical and electrical characteristics. Total mechanical output power P\({}_{\text{out}}\) delivered by the motor is related to its torque T\({}_{\text{out}}\) and rotation speed n by T\({}_{\text{out}}\) = P\({}_{\text{out}}\)/\(\omega\), where \(\omega\) is the angular velocity (rad/s) and n is the rotational speed (rpm). Air gap length L\({}_{\text{g}}\) is crucial for determining a motor's efficiency and performance and can be estimated based on the mechanical power derived from Equations (1) and (2), where \(k_{\text{g}}\) is an empirical coefficient typically between 0.001 to 0.005, D is the stator inner diameter, \(C_{d}\) is the design constant (typically 11-13 for large machines), and L\({}^{\prime}_{\text{g}}\) is the effective length of the stator. \[\text{L}_{\text{g}}=k_{g}\times\text{D} \tag{1}\]  
\[\text{D}\ =\sqrt{\frac{\text{P}_{\text{out}}}{\text{C}_{d}\ \text{L}^{\prime}_{ \text{g}}\text{n}}} \tag{2}\]The number of stator slots \(\mathrm{Q_{s}}\) was calculated based on the winding configuration and pole pairs from Equation (3). \[\mathrm{Q_{s}}=m\ \times p\times q \tag{3}\]  
where \(m\) is the number of phases, \(p\) the number of pole pairs, and \(q\) the number of slots per pole per phase. The electromotive force (EMF) induced in each phase winding is given from Equations (4) and (5) as  
\[E=4.44\ f\times N\times A\times B \tag{4}\]  
\[A=\frac{\pi\ \mathrm{D}^{2}}{4} \tag{5}\]  
where \(f\) is the supply frequency (Hz), \(N\) is the number of turns per phase, \(A\) is the cross-sectional area of the winding, and \(B\) is the flux density in the stator core. * Rotor Design  
Building upon the preliminary design, a rotor configuration was developed, emphasizing factors influencing magnetic flux and mechanical stability. The rotor's diameter and the number of rotor bars were determined similarly to the stator, ensuring the machine operated at the desired efficiency and speed range. The rotor slot shape and material choice, i.e., typically copper for high efficiency, significantly affect a motor's performance. Additionally, the rotor slot geometry directly affects the torque production. For a squirrel-cage rotor, the number of rotor bars should be slightly different from the number of stator slots to reduce harmonic effects. The depth and width of the slots were chosen to balance mechanical strength with electrical performance and were optimized using Finite Element Method (FEM) analysis, covered later in the section on optimization. Rotor slot pitch \(\tau_{r}\) was determined from Equation (6), where \(\mathrm{Q_{r}}\) is the number of rotor slots. \[\tau_{r}=\frac{360^{\mathrm{o}}}{\mathrm{Q_{r}}} \tag{6}\]
* Equivalent Circuit Model  
The next step included the construction of the equivalent circuit, enabling the simulation of electrical behavior under various operating conditions. The equivalent circuit of an induction motor models its electrical performance. The parameters include stator resistance \(\mathrm{R_{s}}\), rotor resistance \(\mathrm{R_{r}}\), stator reactance \(\mathrm{X_{s}}\), and rotor reactance \(\mathrm{X_{r}}\). To determine the values of the equivalent circuit parameters, standard tests were employed. The no-load test was used to evaluate the core losses of the motor, e.g., hysteresis and eddy current losses under unloaded conditions. The locked rotor test, on the other hand, provided insights into the rotor resistance and leakage reactance, both critical for understanding motor starting behavior and torque production. Figure 1 shows the parameters for the equivalent circuit of the three-phase induction motor. The parameters inside the box were acquired through the 2D FE simulations of the indirect tests of the induction motor; they were not constant.