## 4 Stability Analysis of the Powertrain
### Coupled Influence of System Parameters on the Stability of the Powertrain  
The previous results have shown different influencing factors on the stability issue which were separately analyzed. This last subsection proposes a sensitivity analysis consisting of the observation of coupled effects due to various parametric variations on:  
* Inductances;
* Magnetic flux of the magnets;
* Inertia of the generator set (gas turbine--permanent magnet synchronous generator);
* Inertia of the motor set (propeller--permanent magnet synchronous motor). The original parameters of the system being displayed in Table 1, different parameter variations are proposed in Table 4, in which only the unstable speed range based on the root locus analysis is displayed:  
These results indicate various trends related to the speed range during which the system operation is seen as unstable. The rising, resp. falling, arrows are showing the variation (increasing, resp. decreasing) of the unstable speed range with respect to the performances obtained with the base settings (first row). It is possible to summarize the sizing trends which may improve the stability of the AC electric propulsion architecture:  
* Important inductances values;
* Low electromotive forces (image of the magnetic fluxes); \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Addition of** & **Unstable Speed** & **Unstable Speed** & **Unstable Speed** \\
**Inductances (mH)** & **Range (rpm) from** & **Range (rpm) from** & **Range (rpm) from** \\  & **Root Locus** & **Time Simulation** & **Experiments** \\ \hline
0 (base setting) & 260 \(\rightarrow\) 620 (\(\Delta\) = 360) & 290 \(\rightarrow\) 580 & 300 \(\rightarrow\) 560 \\
1 & 240 \(\rightarrow\) 560 (\(\Delta\) = 320) & 270 \(\rightarrow\) 530 & 270 \(\rightarrow\) 530 \\
2 & 230 \(\rightarrow\) 520 (\(\Delta\) = 290) & 250 \(\rightarrow\) 490 & 250 \(\rightarrow\) 500 \\
3 & 210 \(\rightarrow\) 480 (\(\Delta\) = 270) & 240 \(\rightarrow\) 450 & 270 \(\rightarrow\) 490 \\
4 & 200 \(\rightarrow\) 440 (\(\Delta\) = 240) & 230 \(\rightarrow\) 420 & 250 \(\rightarrow\) 470 \\
5 & 190 \(\rightarrow\) 420 (\(\Delta\) = 230) & 220 \(\rightarrow\) 390 & 240 \(\rightarrow\) 450 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results of the different means of study for different series inductances values. Important inertia of the motor set. In order to analyze the coupling effects between parameters and to verify potential antagonistic couplings, the following variations are simultaneously applied to the system:  
10% increase in the generator and the motor inductances;  
10% decrease in the generator and the motor magnetic flux of the magnets;  
10% decrease in the inertia of the generator set;  
10% increase in the inertia of the motor set. The theoretical analysis under these coupled modifications is displayed on the Figure 14 for the root locus and Figure 15 for the time simulation:  
Compared with the results of Figures 7 and 8 related to the base parametric configuration, the stability level is clearly increased with narrow unstable speed range. Indeed, the unstable speed range (seen from the root locus) which was initially between 260 and 620 rpm (\(\Delta=360\) rpm) for the base parametric configuration is now between 240 and 450 rpm (\(\Delta=210\) rpm), showing a great improvement of the stability level of the powertrain. It is even possible to find parametric configurations allowing a \"full stable operation\" of the AC electric propulsion architecture by imposing higher modifications following the same trends as above. For example, the following parametric configuration can be considered: \begin{table} - & 1000 increase in the generator and the motor inductances;  
- & 1000 decrease in the generator and the motor magnetic flux of the magnets;  
- & 1000 increase in the inertia of the generator set;  
- & 1000 increase in the inertia of the motor set.  
\end{table} 
Table 4: Multi-parameter variations and associated results on the stability.