# Synchronous Homopolar Generator without Permanent Magnets for Railway Passenger Cars
## 4 Optimization Results  
The application of the Nelder-Mead method leads to the convergence of the objective function defined by Equation (1) to a certain minimum (Figure 6). In the course of optimization, the SHG average loss (Figure 7a) and the armature current amplitude (Figure 7b) were also reduced significantly. The nonsymmetrized torque ripple, that has the smallest power in the objective function (1), increased during the optimization (Figure 8a). However, the symmetrized torque ripple, which is the resultant one of the SHM as a whole, slightly decreased after the optimization (Figure 8b), which indicates that the torque waveforms of the individual SRSCs are in opposite phase and cancel each other out. Figures 9 and 10 show the geometry and 2D flux density plot of an SRSC before and after optimization.  
Figure 5: Objective function flowchart. The input parameters varied during optimization is shown in Table 2. The objective function is defined by Equation (1).  
Figure 6: Change in the value of the objective function defined by Equation (1) in the course of optimization.  
Figure 8: Change in the SHG parameters in the course of optimization. (**a**) nonsymmetrized torque ripple and (**b**) symmetrized torque ripple.  
Figure 7: Change in the SHG performances in the course of optimization. (**a**) average losses and (**b**) maximum armature current amplitude.  
Figure 9: The cross-section of the SHG initial design and its plot of flux density magnitude (T); areas of extreme saturation (>2T) are colored white: (**a**) at 3450 rpm and (**b**) at 750 rpm.  
Comparison of the SHG parameters before and after optimization, shown in Table 3 allows us to draw the following conclusions:  
1. The losses in operating point 2 reduce by 100% (\(11.56-8.10\))/\(11.56=30\%\);
2. The losses in operating point 1 are much lower than those in point 2, and their increase by \(3.84/2.52=1.5\) times is not so bad. In addition, it is the price for the reduction in the required rectifier power by 100% (\(100.6-67.9\))/\(100.6=32.5\%\), which is achieved by increasing the field weakening angle;
3. However, average losses reduce by 100% (\(7.04-5.97\))/\(7.04=15.2\%\);
4. To demonstrate the ability of the SHG to produce energy at a speed lower than 750 rpm, the 300 rpm mode was calculated. To reduce saturation and losses, the torque equal to 90% of the torque in point 2 was chosen. The output power is 8.03 kW; the input power is 14.42 kW, and the generator efficiency is 56%;
5. After optimization, the maximum symmetrized torque ripple (ripple at the shaft at low speed was reduced by 100% (\(13.1-11.3\))/\(13.1=13.7\%\);
6. Comparison of the initial and the optimized designs shows that as a result of optimization, the area of the stator slots for the winding increased by increasing the height and reducing the thickness of the stator tooth, which leads to a decrease in the armature winding resistance and the DC losses in the winding.  
The original design has a flux density in nonlaminated parts of more than 1.6 T. Due to the introduction of an appropriate multiplier in the optimization function defined by Equation (1), in the final design, the thickness of the stator housing is increased, and the induction does not exceed 1.6 T.  
The rotor sleeve radius is also automatically increased, for the housing and rotor sleeve cross-section areas are equal. The increase in the housing thickness and the stator slot depth leads to a decrease in the airgap radius, which is partially compensated by an increase in the stator stack length. Table 3 compares the SHG characteristics before and after optimization.  
Figure 10: The cross-section of the SHG optimized design and its plot of flux density magnitude (T); areas of extreme saturation (>2T) are colored white: (**a**) at 3450 rpm and (**b**) at 750 rpm.