## 3 The New Power System
### Choice of the Rotational Speeds  
To adapt the system to the present application, the wheel shaft was replaced by a propeller shaft and the transmission was subjected to a first preliminary design, which led to the modification of the transmission ratios to make them more suitable for use with a variable pitch propeller, and thus at a given output shaft speed. Given the configuration chosen for the transmission, the speeds and torque ranges were obtained for the engine and electric motors, respectively, in engine mode and EV series mode when the engine is decoupled from the transmission shaft.  
The second step in the design process was to evaluate the choice of the engine speed on the basis of its performance curve and efficiency map at sea level (Figure 5).  
The wide-open throttle (WOT) power of the engine is 70 kW and is reached at the speed of 7000 rpm, while the WOT torque is maximized at 4700 rpm. Therefore, if the transmission is designed so that the engine runs at 7000 rpm, the mission can be entirely performed in thermal mode and the engine power is fully exploited. Actually, it would be necessary to take into account the reduction in WOT power that is caused by the lower pressure and temperature of the atmosphere at the flight altitude. For naturally aspirated engines, such a reduction can be expressed as [33]:  
\[\frac{BHP(z)}{BHP(0)}=1.13\sigma(z)-0.13 \tag{1}\]  
where \(z\) is the flight altitude, \(BHP\) is the engine WOT brake horsepower, and \"\(\sigma(z)\)\" is the ratio between air density at \(z\) and air density at sea level (z = 0).  
Figure 8 shows the percentage of loss of engine WOT power as a function of the flight altitude. Keeping in mind the mission in Figure 3, a 10% loss in the engine is obtained for the specific application.  
However, in order to reduce fuel consumption, it could be suitable to make the engine work at its best efficiency point (optimal operating point, OOP), but in this case, the sea level power should be only 30 kW. To compromise between performance and efficiency, a rotational speed of 4590 rpm was chosen for the Wankel engine. The relevant operating points of the engine considered in this investigation are summarized in Table 1.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Operating Point** & **Speed (rpm)** & **Torque (Nm)** & **Power (kW)** & **Efficiency (\%)** \\ \hline Max power (WOT) & 7000 & 95.5 & 70.0 & 0.220 \\ Max torque (WOT) & 4700 & 109.0 & 53.7 & 0.230 \\ Best efficiency (OOP) & 3000 & 95.0 & 29.8 & 0.240 \\ WOT and OOL @ selected speed & 4590 & 107.7 & 51.8 & 0.235 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Relevant engine working points.  
Figure 7: Transmission scheme of the HHS.  
As for the electric machines, the speeds of EM1 and EM2 were chosen with a trial-and-error procedure considering a constant output speed of 2000 rpm. The geometrical and functional limits of the proposed power split device and the range of feasibility of permanent magnet machines were taken into account in the procedure. The results of the analysis led to the choice of 8570 rpm for EM2 and 7000 rpm for EM1. The number of teeth and the transmission ratio of the original and modified systems are reported in Table 2.