# Topology Comparison Study of Five-Phase Wound-Field Doubly Salient Fault Tolerant Generators
## 3 Comparative Study and Simulation of Five-Phase WFDSG
### WFDSG Design Constraints  
Although the new five-phase WFDSG differs from the conventional machine in structure and principle, the WFDSG still conforms to the general law of electromechanical energy conversion, and the basic parameters of the WFDSG can still be analysed by the electromagnetic formula of the electric machine. Let the output capacity of the machine be:  
\[P=mui. \tag{7}\]  
Here, \(m\) is the number of phases, \(u\) and \(i\) are the voltage and current of each phase. Ignoring the effect of the saturation effect, the voltage waveform of the five-phase WFDSG can be regarded as a square wave, and its peak value \(u_{\rm max}\) can be calculated as:  
\[u_{\rm max}\approx N_{\rm p}(\phi_{\rm max}-\phi_{\rm min})\frac{np_{\rm r}}{ 60}\frac{360}{\theta}. \tag{8}\]  
Here, \(N_{\rm p}\) is the total turn number of the series connected phase turns, \(n\) is the number of stator poles and rotor poles overlapping, \(\phi_{  
Substituting Equation (8) and \(k_{\rm d}\) into Equation (7), and then Equation (7) can be calculated as:  
\[u_{\rm max}\approx\frac{\pi k_{\rm d}DN_{\rm p}LB_{\rm g}n}{60}. \tag{11}\]  
It can be seen from Equation (9) that by adjusting the magnitude of the excitation current, the air gap magnetic field can be adjusted to achieve the purpose of keeping the output voltage stable. At the same time, since there is no pole number and pole arc coefficient in Equation (9), changing the pole number and pole arc coefficient has no effect on the peak value of the output voltage, under the condition of neglecting magnetic flux leakage and magnetic saturation. This is because although a large number of poles increase the frequency, it also reduces the magnetic flux per pole. Transform Equation (9), and the formula of the armature winding turns can be calculated as:  
\[N_{\rm p}\approx\frac{60u_{\rm max}}{\pi k_{\rm d}DLB_{\rm g}n}. \tag{12}\]  
Therefore, it can be seen that in the case where the external dimensions of the WFDSG and the magnetic gap of the air gap are constant, the output voltage of different WFDSCs are the same if they have the same total turns number of the armature windings in each phase. The design requirement of the field winding of the five-phase WFDSG is to sufficiently provide enough field magnetizing. In the case of instauration, the magnetic force of the core is relatively small. Therefore, according to the second law of Kirchhoff and the size of the air gap, the excitation magnetic potential can be calculated, and then the required excitation slot area can be calculated according to the winding current density. The design method of the armature winding and the field winding given above can be used to calculate the slot fill factor of the WFDSG core. The design optimization of the WFDSG parameter should consider the following constraints:  
\[ \begin{cases}\text{Split ratio}:&\gamma=\frac{D}{D_{0}}<1\\ \text{Diameter}:&D_{0}-D=2(h_{\rm c}+h_{\rm e})\\ \text{Second air gap}:&h_{\rm c}>6 nm\\ \text{Stator yoke}:&h_{\rm e}\geq\frac{\pi D_{\rm 2s}}{h_{\rm e}}=b_{\rm c} \\ \text{Field slot fill factor}:&\frac{N_{\rm p}d_{\rm 2}^{2}+2N_{\rm p}d_{\rm 2}^{2}}{ \frac{N_{\rm p}d_{\rm 2}^{2}+2N_{\rm p}d_{\rm 2}^{2}}{360}\pi(D_{\rm 0}-2h_{\rm c}-h_{\rm c })h_{\rm c}}\leq 0.6\\ \text{Armature slot fill factor}:&\frac{N_{\rm p}d_{\rm 2}^{2}}{360}\pi(D_{\rm 0 }-2h_{\rm c}-h_{\rm c})h_{\rm c}\leq 0.5\end{cases}  \tag{13}\]  
where, \(\gamma\) is the split ratio of the WFDSG, \(D_{0}\) and \(D\) are the out diameter and air gap diameter, \(h_{\rm e}\) and \(h_{\rm c}\) are the teeth height and yoke height of the stator, and \(h_{\rm c}\) is the teeth width of the stator.