# Topology Comparison Study of Five-Phase Wound-Field Doubly Salient Fault Tolerant Generators
## 3 Comparative Study and Simulation of Five-Phase WFDSG
### Rotor Pole Optimization Considering the Fringing Effect  
Ignoring the core fringing effects, the peak and valley lengths of the inductance waveforms of each phase should be equal when the stator pole arc coefficient \(\alpha_{\rm r}=0.5\). According to the inductance and back EMF characteristics of the five-phase WFDSG, the back EMFs of the machine are also approximately square waves. However, due to the core fringing effect, the magnetic flux is conducting before the stator and rotor poles start to slip together, as shown in Figure 3. Since the principle is similar to the actual length of the iron core in the machine design, it can be used to refer to the definition of the equivalent length of the core in the machine design. Here, let the rotor pole arc equivalent length \(l_{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{ \rm{\rm{\rm{\rm{ }}}}}}}}}}}}}}}}}}\) be the performance functional pole arc length, and it is larger than the rotor pole arc actual length \(l_{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{{\rm{{\rm{\rm{{\rm{\rm{{\rm{{\rm{{\rm{ }}}}}}}}}}}}}}}}}}}\), as shown in Figure 3.  
In order to obtain the rotor pole arc actual length that has the best output performance of the WFDSG, let:  
\[l_{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{{\rm{{\rm{{\rm{ \rm{\rm{{{\rm{{{}}}}}}}}}}}}}}}}}}}}=l_{\rm{\rm{\rm{\rm{\rm{ \rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{{\rm{{\rm{\rm{{\rm{{\rm{{{}}}}}}}}}}}}}}}}}}}}}}-z\delta\,} \tag{14}\]  
where \(z\) is the unknown number to be solved, and \(\delta\) is the air gap length. For a five-phase WFDSG, when the peak and valley lengths of the inductance waveforms of each phase are equal, \(\alpha_{\rm r}=0.5\), the ideal value of the equivalent arc length of the rotor should be:  
\[l_{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{{ \rm{\rm{\rm{{\rm{{{}}}}}}}}}}}}}}}}}}}}=0.5\frac{\pi D}{p_{r}}= \frac{\pi D}{2p_{r}}. \tag{15}\]  
In order to obtain the optimized z in Equation (12), the method of intersecting two rotor cores in the simulation software is used to establish a 20/16-pole five-phase WFDSG optimisation model. When z is selected from 0 to 5, the flux linkage, no-load EMF and cogging torque of phase A were  
Figure 3: Rotor pole equivalent length. Figure 2: The simulated electromotive force (EMF) and flux linkage. (**a**) Phase EMF of 30/24-pole. (**b**) Phase EMF of 20/16-pole. (**c**) Phase flux linkage of 30/24-pole. (**d**) Phase flux linkage of 20/16-pole. simulated [34]. And the waveform diagrams and harmonic diagram were obtained as shown in Figure 4.  
It can be seen that the arc length of the rotor has a direct influence on the peak and valley length of the flux linkage, as well as affects the induced EMF waveforms. When the arc length of the rotor changes within the above range, the amplitudes of no-load induced EMF are almost equal. As z gradually decreases from 4 to 0, the rotor pole actual length gradually approaches the ideal pole arc length in Equation (13), the fundamental wave content of the no-load EMF and the flux linkage gradually decrease. On the other hand, if \(z\) is increased to 5, the rotor pole length is smaller than the ideal pole arc length in Equation (13), and fundamental wave content of the no-load EMF and the flux linkage are also smaller than that of \(z=4\). In summary, when \(z=4\), the fundamental wave content of each phase flux and no-load EMF is the largest. In the cogging torque waveform showed in Figure 4c, when \(z\) gradually decreases from 4 to 0, the cogging torque increases correspondingly. When \(z=5\), the cogging torque is also relatively large. The minimum cogging torque is achieved when \(z=4\). This is also consistent with the above fundamental wave content analysis. Therefore, when the rotor pole arc length \(l_{\text{rp}}\) is \(4\delta\) narrower than the ideal pole arc length, the cogging torque of the five-phase WFDSG is the smallest, the flux linkage and no-load EMF harmonic content are also the smallest.