# Design of a Rotary Transformer for Installations on Large Shafts
## 4 Design of the Transformer
### Genetic Optimization  
Starting from the result obtained from the grid search algorithm, a multi-objective genetic optimization was set to improve the performance of the best candidate. The genetic optimization was performed in order to converge faster than the brute-force approach to a solution for the transformer considering the large number of constraints that are imposed on this design. The goal was to obtain the best turn ratio, not too far from one, as well as to refine the geometrical dimensions of the core in order to transfer 3.6 kW on the secondary while minimizing the weight of the rotary transformer. Regarding the number of turns, \(N_{1}\) and \(N_{2}\) were now relatively free to vary since the selected load did not have strict voltage limits, different than in [28, 29], in which the transformation ratio was imposed. A 30% maximum variation of the turn ratio was considered. Litz wires were no longer considered in this investigation, since the operational frequency was not too high and therefore costs could be reduced. As a matter of fact, investigations between these two optimizations highlight that skin effect should not be an issue if the wires are wisely sized. Indeed, conductors can be connected in parallel inside each slot (namely, a sort of Litz wire with a low number of strands having greater cross sections) to improve the filling factor and limit the resistance of the associated winding. Moreover, the same limit on the current density foreseen for the grid-search optimization was set. All these goals were achieved in MATLAB(r), finding the Pareto front of three fitness functions using the genetic algorithm. These three functions to be minimized were the overall weight of the device, \(|P_{2}-3.6\,\mathrm{kW}|\) and \(|j_{1}-10\,\mathrm{A/mm^{2}}|\). It is important to note that \(j_{1}\) was chosen because the primary current is always higher than the secondary. Moreover, in view of a practical realization of a prototype of the device, the two cores were constrained to be equal to simplify their manufacturing process. The size of the flux path in the whole cross section was set constant (as suggested in [2]), that is, \(w_{\mathrm{c1}}=w_{\mathrm{c2}}=w_{\mathrm{c}}\), and the width of the winding  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Grid-Search** & **Genetic Algorithm** & **C-Ferries** \\ \hline \(w_{\mathrm{c}}\) & 4.20 mm & 4.04 mm & 4.04 mm \\ \hline \(w_{\mathrm{c1}}\) & 5.40 mm & 4.04 mm & 4.04 mm \\ \hline \(w_{\mathrm{c2}}\) & 4.78 mm & 4.04 mm & 4.04 mm \\ \hline \(w_{\mathrm{w}}\) & 2.97 mm & 2.05 mm & 2.05 mm \\ \hline \(P_{2}\) & 2.70 kW & 3.72 kW & 3.76 kW \\ \hline \(L_{1}\) & 130.61 μH & 306.46 μH & 283.04 μH \\ \hline \(L_{2}\) & 130.70 μH & 427.83 μH & 395.87 μH \\ \hline \(M\) & 125.89 μH & 351.55 μH & 325.59 μH \\ \hline \(N_{1}\) & 7 & 11 & 11 \\ \hline \(N_{2}\) & 7 & 13 & 13 \\ \hline
**Weight** & 0.91 kg & 0.66 kg & 0.55 kg \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison between the results obtained in the three optimization procedures.  
slots \(w_{\rm w}\) was considered a variable, as well. Therefore, the following ranges of variations were considered:  
* \(2\,\mathrm{mm}\leq w_{\rm w}\leq 6\,\mathrm{mm}\);
* \(4\,\mathrm{mm}\leq w_{\rm c}\leq 6\,\mathrm{mm}\)
* \(5\leq N_{1}\leq 15\)
* \(5\leq N_{2}\leq 15\)  
Furthermore, height of the cores, air gap width and shaft radius were set constant, as shown in the previous subsection. The outcome of this procedure was a transformer with the characteristics reported in Table 1. The genetic optimization ran in about 31,705 s on a desktop PC with Intel(r) Core(tm) i7-8700 processor at 3.20 GHz and with 16 GB RAM.