## 2 Machine Configuration and Operating Principle
### Working Harmonic Selection  
The amplitude of the harmonics in type B is lower than type A with the same \(i\) under the flux modulation effect. Therefore, to obtain higher back electromotive force (EMF) and associated electromagnetic torque, the number of the pair of winding poles should be designed according to the harmonic order of type A. According to (8), the highest value of \(F_{ri}\) can be obtained when \(i=1\). Therefore, the pair number of the winding should be designed as follows:  
\[p_{a}=p_{r} \tag{27}\]  
Under the abovementioned winding design, all the harmonics in types A and B with \(i=1\) are utilized as working harmonics. As all three types of working harmonics have the same frequency, the back EMF of the three types of harmonics can be added together.  
According to (10), the highest value of \(\Lambda_{j}\) can be obtained when \(j=1\). Therefore, the harmonics with \(j=1\) in type B are used as the main working harmonics.  
The slot angles of types A, B1, and B2 harmonics can be calculated as follows:  
\[\alpha_{A}=\frac{2\pi p_{r}}{Z} \tag{28}\]  
\[\alpha_{B1}=\frac{2\pi(p_{r}+p_{m})}{Z} \tag{29}\]  
\[\alpha_{B2}=\frac{2\pi|p_{r}-  
All three types of working harmonics have the same distribution factor and pitch factor, i.e.,  
\[\left\{\begin{array}{l}k_{dA}=k_{dB1}=k_{dB2}=k_{d}\\ k_{pA}=k_{pB1}=k_{pB2}=k_{p}\end{array}\right. \tag{34}\]  
The coil pitch of types A and B1 is opposite from type B2, i.e.,  
\[\left\{\begin{array}{l}y_{A}=y_{B1}=y\\ y_{B2}=-y\end{array}\right. \tag{35}\]  
_i. Flux controlling principle_  
Total back EMF can be derived as the vector sum of the back EMF of types A, B1, and B2. From (24)-(26), we can conclude the vector graph of the back EMF, as shown in Figure 7a. Vector \(\mathbf{E_{A}}\) is fixed, whereas \(\mathbf{E_{B1}}\) and \(\mathbf{E_{B2}}\) can rotate in the opposite direction. The total back EMF can change along the blue oval-shaped trajectory. When the electrical initial phase angle of the modulator \(p_{m}\phi=0^{\circ}\), the components of types A, B1, and B2 have the same direction and the total back EMF reaches the maximum at this position, as shown in Figure 7b. When \(p_{m}\phi=180^{\circ}\), both types B1 and B2 have the opposite direction from type A, and the total back EMF reaches the minimum at this position, as shown in Figure 7c. Therefore, although types B1 and B2 have different rotation directions, both type B components positively contribute to the control of the back EMF.  
By adjusting the electrical initial phase angle of the modulator \(p_{m}\phi\) from \(0^{\circ}\) to \(180^{\circ}\), the total back EMF can vary with the range from \(|E_{mA}-E_{mB1}-E_{mB2}|\) to \((E_{mA}+E_{mB1}+E_{mB2})\). Therefore, the total back EMF amplitude can be effectively controlled.  
Figure 7: Back EMF vector graph. (**a**) Showing diagram; (**b**) maximum total back EMF position (\(p_{m}\phi=0^{\circ}\)); (**c**) minimum total back EMF position (\(p_{m}\phi=180^{\circ}\) ).  
From (25) and (26), it should be observed that the amplitude of \(\mathbf{E_{B1}}\) is significantly smaller than \(\mathbf{E_{B2}}\). As the rotating speed of \(\mathbf{E_{B1}}\) is \(\left(\frac{p_{r}+p_{m}}{p_{r}-p_{m}}\right)\) times higher than \(\mathbf{E_{B2}}\); therefore, \(\mathbf{E_{B2}}\) plays a major role in the control of the back LMF. To obtain the widest control range of the back EMF, the amplitude of \(\mathbf{E_{B2}}\) should be close to \(\mathbf{E_{A}}\). From (24) and (26), we could observe that \(B_{mB2}\) is smaller than \(B_{mA}\), as \(\Lambda_{j}\) is always smaller than \(2\Lambda_{0}\). Therefore, \(\omega_{B2}\) should be larger than \(\omega_{A}\), and the pole pair number of rotor should follow:  
\[p_{r}>\frac{Z}{2} \tag{36}\]