# Optimization Design and Control of Six-Phase Switched
## 2 The Phenomenon of Current Asymmetry
### Magnetic Circuit Analysis  
The stator yokes between phases A and B are selected as the analysis objects. The polarities of adjacent poles between phases A and B could be the same as NN or the opposite NS, which correspond to the magnetic field distributions between phases A and B, and phases B and C in Figure 1b, respectively. Assuming that the magnetic flux leakage is ignored, the magnetic circuits under two kinds of magnetic pole arrangements are presented in Figure 3.  
In Figure 3, \(R_{\text{SY}}\) and \(R_{\text{RY}}\) are the reluctances of stator yoke and rotor yoke, respectively, \(R_{\text{A1}}\), \(R_{\text{B1}}\), and \(R_{\text{B2}}\) are the sum of the reluctances stator teeth and rotor teeth corresponding to poles \(A_{1}\), \(B_{1}\), and \(B_{2}\) and their aligned rotor poles, respectively, \(R_{\text{SA}}\) and \(R_{\text{SB}}\) are the air gap reluctances of phase A and phase B, respectively, \(F_{\text{A1}}\), \(F_{\text{B1}}\), and \(F_{\text{B2}}\) are the magnetomotive force excited by the windings on poles \(A_{1}\), \(B_{1}\), and \(B_{2}\), respectively, and \(\phi\) is the magnetic flux. Since SRM is the type of symmetric motor and each stator pole is excited with the same current condition, the magnetomotive forces shown in the two figures in Figure 3 are equal, which leads to  
\[\frac{\phi_{\text{NN}}}{\phi_{\text{NS}}}=\frac{R_{\text{A1}}+R_{\text{B1}}+R_ {\text{SA}}+R_{\text{SB}}+R_{\text{SY}}+R_{\text{RY}}}{R_{\text{A1}}+R_{\text {B2}}+R_{\text{SA}}+R_{\text{SB}}+5R_{\text{SY}}+4R_{\text{RY}}} \tag{1}\]  
Assume that the magnetic flux density of the stator yoke is B = 1.5 T, and the magnetic permeability of the silicon steel sheet is 1500 times that of the vacuum. Based on the parameters in Table 1. The relationships between the other reluctances with the airgap reluctance \(R_{\text{SA}}\) can be represented by  
\[R_{\text{SA}}=R_{\text{SB}}/2=115R_{\text{A1}}=15R_{\text{SY}}=22R_{\text{RY}} \tag{2}\]  
Furthermore, based on (1) and (2), it can be deduced that  
\[\phi_{\text{NN}}/\phi_{\text{NS}}  
It is acknowledged that with the increase in the magnetic flux density, the change rate of the magnetic field gradually decreases. From Formulas (1)-(4), the excitation current rises faster with NS arrangement than that with NN arrangement, which leads to the asymmetric phase currents in six-phase SRM with NNSS winding connection. Based on the above analysis, it can be concluded that when the windings are connected in forward series with the magnetic pole distribution alternately arranged by NNSS, the excitation currents are unequal. In other words, the current asymmetry is caused by the asymmetric magnetic flux density of the yokes; therefore, the asymmetric phase currents can be relieved by optimizing the stator yoke width or selecting a suitable control scheme. Set the total magnetic resistance except for the stator yoke as \(\Sigma R\) in Figure 3b, and then the magnetic flux in the circuit can be expressed as  
\[\left\{ \begin{array}{l}\phi_{\text{NS}}=BS_{\text{SY}}=\frac{N_{\text{ph}}i _{\text{A}}+N_{\text{ph}}i_{\text{B}}}{R_{\text{SY}}+2K}\\ R_{\text{SY}}=\frac{I_{\text{s}}}{\mu_{\text{Fe}}S_{\text{SY}}}\end{array} \right. \tag{5}\]  
where \(B\), \(S_{\text{sy}}\), \(N_{\text{ph}}\), \(i_{\text{B}}\), \(I_{\text{a}}\), and \(\mu_{\text{Fe}}\) represent magnetic flux density, the cross-sectional area of stator yoke, current of phase B, lamination length, and permeability of iron, respectively. Equation (5) can be rewritten as  
\[B=\frac{N_{\text{ph}}i_{\text{A}}+N_{\text{ph}}i_{\text{B}}}{\left(\frac{I_{ \text{a}}}{\mu_{\text{Fe}}}+S_{\text{SY}}\Sigma R\right)} \tag{6}\]  
Equation (6) shows that under the same number of turns and stack length, the magnetic flux density can be reduced in two ways, i.e., reduction in the magnetomotive force and increase in the stator yoke area. The first way can be realized by shortening the overlapping angle of conduction between adjacent phases or reducing the conduction width of each phase.