# Topology Comparison Study of Five-Phase Wound-Field Doubly Salient Fault Tolerant Generators
## 2 Topology and Principle of the Five-Phase WFDSG  
In the conventional five-phase WFDSGs, each excitation coil is wound across five stator poles. The stator pole number should be an even multiple of the phase number 5, that is, the minimum unit machine is a 10/8 pole structure. Literature [32] has pointed out that the traditional five-phase WFDSGhas an amplitude difference in the inductance of each phase, and this asymmetry in the inductance of each phase will result in asymmetrical phases, which will increase the unbalanced phase load.  
Formula for ratio of stator pole number to rotor pole number of five-phase WFDSG [32]:  
\[\frac{p_{s}}{p_{r}}=\frac{5}{4}\;\text{or}\;\frac{5}{6}\;. \tag{1}\]  
The symmetry of each phase winding can be achieved by changing the number of poles wound by the field winding. When each field winding is wound across 2 stator poles, the number of stator poles is at least 20. If each field winding is wound across 3 stator poles, the stator pole number is at least 30. For the WFDSG with inner rotor, the number of rotor poles is usually less than the number of stator poles. Therefore, two kinds of five-phase WFDSGs can be proposed, one has 30 stator pole and 24 rotor poles, which is summarized as 30/24-pole, and the other is structured with 30/24-pole, as shown in Figure 1.  
When the excitation current of the five-phase WFDSG is turned on, the excitation magnetic field is established inside the machine. When the five-phase WFDSG armature winding and the excitation winding are all conducted with currents, the flux linkage of the armature winding and the excitation winding can be expressed by the following formula:  
\[[\psi]=[\text{L}][\text{I}], \tag{2}\]  
where the matrix \([\psi]\), \([\text{L}]\), and \([\text{I}]\) represent the flux linkage, inductance, and current matrix, respectively, and the subscripts a, b, c, d, e, and f correspond to phase A, phase B, phase C, phase D, phase E, and field winding.  
\[[\psi]=[\psi_{\text{a}},\psi_{\text{b}},\psi_{\text{c}},\psi_{\text{d}},\psi_ {\text{e}},\psi_{\text{f}}]^{\prime} \tag{3}\]  
\[[L]=\left[\begin{array}{ccccc}L_{\text{a}}&L_{\text{ab}}&L_{\text{ac}}&L_{ \text{ad}}&L_{\text{ae}}&L_{\text{af}}\\ L_{\text{ba}}&L_{\text{b}}&L_{\text{bc}}&L_{\text{bd}}&L_{\text{be}}&L_{\text {bf}}\\ L_{\text{ca}}&L_{\text{cb}}&L_{\text{c}}&L_{\text{cd}}&L_{\text{ce}}&L_{\text {cf}}\\ L_{\text{da}}&L_{\text{db}}&L_{\text{dc}}&L_{\text{d}}&L_{\text{de}}&L_{\text {df}}\\ L_{\text{ea}}&L_{\text{eb}}&L_{\text{ce}}&L_{\text{ed}}&L_{\text{e}}&L_{\text {ef}}\\ L_{\text{fa}}&L_{\text{fb}}&L_{\text{fc}}&L_{\text{fd}}&L_{\text{fe}}&L_{\text {f}}\\ \end{array}\right] \tag{4}\]  
\[[\text{I}]=[i_{\text{a}},i_{\text{b}},i_{\text{c}},i_{\text{d}},i_{\text{e}},i_{\text{f}}]^{\prime} \tag{5}\]  
Figure 1: Two topology of the five-phase wound-field doubly salient generators (WFDSGs). (a) 30/24-pole; (b) 20/16-pole.  
Because the armature windings of the WFDSG are concentrated windings wound in the same direction, and the numbers of phase turns are much smaller than the excitation winding, the mutual inductance flux linkage is small, and the mutual inductance between the armature windings can be neglected, and the flux linkage equation can be rewritten as:  
\[\left[\begin{array}{c}\psi_{\rm a}\\ \psi_{\rm b}\\ \psi_{\rm c}\\ \psi_{\rm d}\\ \psi_{\rm e}\\ \psi_{\rm f}\end{array}\right]=\left[\begin{array}{cccccc}L_{\rm a}&0&0&0& 0&L_{\rm af}\\ 0&L_{\rm b}&0&0&0&L_{\rm bf}\\ 0&0&L_{\rm c}&0&0&L_{\rm cf}\\ 0&0&0&L_{\rm d}&0&L_{\rm df}\\ 0&0&0&0&L_{\rm e}&L_{\rm ef}\\ L_{\rm fa}&L_{\rm fb}&L_{\rm fc}&L_{\rm fd}&L_{\rm fe}&L_{\rm f}\\ \end{array}\right] \tag{6}\]  
From Equation (4), it can be seen that the mutual inductance between the five-phase windings of the generator is small, the five-phase windings are relatively independent, and have strong isolation characteristics. The failure of one or more phases has little effect on the remaining phases. Therefore, the five phases of the generator can simultaneously output current independently and have strong fault tolerance.