# Design Optimization of the Magnet-Free Synchronous Homopolar Motor of a Subway Train
## 7 Optimization Results  
Figure 6 shows the change of the electrical losses and current magnitude during optimization stage. Figure 7 depicts the simultaneous change of the symmetrized and non-symmetrized torque ripple. Figure 8 illustrates the change of the cost function value along the optimization. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Initial Design** & **Optimized Design** \\ \hline Housing thickness \(h\), mm & 20 & 25.7 \\ \hline Total stator stacks length \(L_{attar}\), mm & 210 & 228 \\ \hline Stator slot depth, \(h_{p}\), mm & 36 & 30.6 \\ \hline Stator slot width, \(b_{p}\), mm & 7 & 7.5 \\ \hline Airgap width \(\delta\), mm & 2 & 3.0 \\ \hline Rotor slot thickness, \(\alpha_{1}\), deg & 36 & 36.7 \\ \hline Rotor slot thickness, \(\alpha_{2}\), deg & 63 & 64.3 \\ \hline Angles of field weakening at operating & & \\ points 1,2,3,4 electrical radians & & 0.5; 0.8; 0.4; 0.3 & 0.64; 0.91; 0.34; 0.17 \\ \hline Current ratio & 5 & 5.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Geometric parameters varied during the optimization. Figure 6: Change of the motor parameters during optimization. (**a**) Average losses; (**b**) maximum magnitude of the power winding current. The following conclusions can be made by comparing motor parameters from Table 4 before and after optimization:  
1. In the initial design, the magnetic flux density in the housing and the sleeve exceeds the chosen level of 1.6 T. After the optimization, the restriction of this value is fulfilled. 2. Average losses were reduced by (12.43 \(-\) 12.03)/12.43 = 3.2%; \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Parameter** & \multicolumn{6}{c}{**Initial Design**} & \multicolumn{6}{c}{**Optimized Design**} \\ \hline
**Operating Point, \(i\)** & **1** & **2** & **3** & **4** & **5** & **1** & **2** & **3** & **4** & **5** \\ \hline Rotational speed \(n\), rpm & 4280 & 4280 & 2854 & 0 & 1427 & 4280 & 4280 & 2854 & 0 & 1427 \\ \hline Amplitude of the armature phase current \(l_{arm}\), A & 316 & 554 & 642 & 647 & 651 & 311 & 541 & 547 & 542 & 545 \\ \hline Efficiency, \% & 94.3 & 94.9 & 95.4 & 0.0 & 94.3 & 94.8 & 95.0 & 95.6 & 0.0 & 94.2 \\ \hline Output mechanical power P\({}_{\rm mech}\), kW & 185.3 & \(-\)370.6 & \(-\)370.6 & 0.0 & 185.3 & \(185.3\) & \(-\)370.6 & \(-\)370.6 & 0.0 & 185.3 \\ \hline Input electrical power, kW & 196.6 & \(-\)351.7 & \(-\)353.7 & 8.3 & 196.4 & 195.4 & \(-\)352.2 & \(-\)354.4 & 8.9 & 196.8 \\ \hline Mechanical loss, kW & 3.55 & 3.55 & 1.06 & 0.00 & 0.14 & 3.55 & 3.55 & 1.06 & 0.00 & 0.14 \\ \hline Armature DC copper loss, kW & 1.79 & 5.51 & 7.39 & 7.50 & 7.60 & 2.48 & 7.53 & 7.67 & 7.55 & 7.62 \\ \hline Armature eddy-current copper loss, kW & 1.15 & 3.96 & 3.25 & 0.00 & 0.83 & 0.59 & 2.25 & 2.16 & 0.00 & 0.60 \\ \hline Stator lamination loss, kW & 4.14 & 4.63 & 3.99 & 0.00 & 1.68 & 2.87 & 3.39 & 3.64 & 0.00 & 1.65 \\ \hline Rotor lamination loss, kW & 0.48 & 0.62 & 0.39 & 0.00 & 0.10 & 0.19 & 0.30 & 0.22 & 0.00 & 0.06 \\ \hline Excitation copper loss, kW & 0.18 & 0.59 & 0.79 & 0.79 & 0.79 & 0.46 & 1.41 & 1.44 & 1.40 & 1.40 \\ \hline Total loss, kW & 11.30 & 18.86 & 16.88 & 8.29 & 11.14 & 10.14 & 18.43 & 16.19 & 8.95 & 11.47 \\ \hline Average losses according to Formula (5) & \multicolumn{6}{c}{12.43} & \multicolumn{6}{c}{12.02} \\ \hline Number of turns in armature winding & \multicolumn{6}{c}{5.93} & \multicolumn{6}{c}{6.68} \\ \hline Power factor & 0.986 & \(-\)0.996 & \(-\)0.924 & 1.000 & 0.920 & 0.994 & \(-\)1.000 & \(-\)0.927 & 1.000 & 0.905 \\ \hline Line-to-line voltage amplitude \(V_{a}\), V. & 728 & 749 & 699 & 14 & 383 & 718 & 755 & 825 & 16 & 468 \\ \hline Nonsymmetrized Torque ripple, \% & 48.9 & 33.7 & 23.4 & 23.4 & 23.4 & 33.9 & 25.9 & 18.8 & 19.0 & 19.0 \\ \hline Symmetrized torque ripple, \% & 9.59 & 7.40 & 4.90 & 5.04 & 5.03 & 7.21 & 5.47 & 3.77 & 3.91 & 3.91 \\ \hline Magnetic flux density in the housing and the sleeve, T & 1.13 & 1.43 & 1.70 & 1.73 & 1.73 & 0.95 & 1.33 & 1.56 & 1.60 & 1.60 \\ \hline \hline \end{tabular}
\end{table}