## 2 Machine Configuration and Operating Principle
### Flux Modulation Principles  
The parameter definition of the analytical model is shown in Figure 2. Suppose that the number of stator slots, the number of pole pair of the rotor, and the number of pole pair of modulators are \(Z\), \(p_{r}\), and \(p_{m}\). When neglecting the magnetic resistance of the iron core, the slotting effect of the stator core, and the flux leakage, the magnetic circuit of one pole can be given in Figure 3 [20]. The magnetomotive force (MMF) of one piece of PM can be expressed as follows:  
\[F_{pm}=H_{c}h_{pm} \tag{1}\]  
Figure 1: Configuration of the mechanical flux-weakening design of the PM generating system. Figure 2: Parameter definition of the analytical model. where \(H_{c}\) is the coercivity of the PM, and \(h_{pm}\) is the thickness of the PM, as shown in Figure 2. The resistance of the PM airgap of one pole and the outer airgap of one pole can be given as follows:  
\[R_{pm}=\frac{h_{pm}}{\mu_{0}\mu_{pm}L_{pm}l_{sk}} \tag{2}\]  
\[R_{gi}=\frac{2g_{i}}{\mu_{0}\theta_{r}g_{sl}} \tag{3}\]  
\[R_{go}=\frac{2g_{o}}{\mu_{0}\theta_{r}g_{sl}} \tag{4}\]  
where \(\mu_{0}\) is the magnetic permeability of the vacuum, \(\mu_{pm}\) is the relative magnetic permeability of the PM material, \(g_{o}\) and \(g_{i}\) denote the length of the inner and outer airgap, \(\theta_{r}\) is the thickness of the iron core of one pole, \(r_{g}\) is the radius of the airgap, and \(l_{sk}\) is the stack length. The flux of one pole can be calculated as follows:  
\[\Phi_{m}=\frac{F_{pm}}{R_{pm}+2R_{gi}+2R_{go}} \tag{5}\]  
The amplitude of one pole's MMF in the airgap can be expressed as follows:  
\[F_{m}=2\Phi_{m}\big{(}R_{gi}+R_{go}\big{)} \tag{6}\]  
The waveform of the rotor in the airgap is shown in Figure 4, which can be expressed as follows:  
\[F_{pm}=H_{c}h_{pm} \tag{1}\]0  
where \(F_{ri}\) is the magnitude of the i-th rotor PM MMF component, which can be further derived as follows:  
\[F_{pm}=H_{c}h_{pm} \tag{1}\]1  
\[F_{pm}=H_{c}h_{pm} \tag{1}\]2 MMF waveform of the rotor PMs. \[F_{pm}=H_{c}h_{pm} \tag{1}\]3 Magnetic circuit of one pole. The permeance of the modulator is presented in Figure 5, which can be expressed as follows:  
\[F_{pm}=H_{c}h_{pm} \tag{1}\]4  
where \(\phi\) is the mechanical initial phase angle of the modulator, \(\Lambda_{0}\) is the magnitude of the modulators' average permeance component, and \(\Lambda_{j}\) is the magnitude of the j-th permeance component, which can be expressed as follows:  
\[F_{pm}=H_{c}h_{pm} \tag{1}\]5  
The airgap flux density produced by the rotor PMs and the modulator teeth can be deduced by multiplying (7) and (9) and is expressed as follows:  
\[F_{pm}=H_{c}h_{pm} \tag{1}\]6  
where  
\[F_{pm}=H_{c}h_{pm} \tag{1}\]7  
\[F_{pm}=H_{c}h_{pm} \tag{1}\]8  
\[F_{pm}=H_{c}h_{pm} \tag{1}\]9  
For convenience in further analysis, the coefficients of the components from (12) to (14) are sorted and listed in Figure 1:0 Harmonic components can be categorized into the following three types:  
1. Type A components are unmodulated stationary components. 2. Type B1 components are modulated components that rotate in the same direction as type A and can be adjusted mechanically. 3. Type B2 components are modulated components that rotate in the opposite direction from type A and can be adjusted mechanically. The flux linkage of one phase can be calculated as follows:  
Figure 1:1 \begin{table}
\begin{tabular}{c c c c c c} \hline
**Type** & **Components** & **Order** & **Speed** & **Frequency** & **Elec. Initial Phase Angle** \\ \hline A & \(B_{r1}\) & \(ip_{r}\) & \(\Omega\) & \(ip_{r}\Omega\) & 0 \\ B1 & \(B_{r2}\) & \(ip_{r}+jp_{m}\) & \(\frac{jp_{r}\Omega}{jp_{m}+ip_{r}}\) & \(ip_{r}\Omega\) & \(jp_{m}\phi\) \\ B2 & \(B_{r3}\) & \(|ip_{r}-jp_{m}|\) & \(-\frac{jp_{r}\Omega}{|jp_{r}-jp_{m}|}\) & \(ip_{r}\Omega\) & \(jp_{m}\phi\) \\ \hline \end{tabular}
\end{table} 
Table 1: Coefficients of the PM harmonic components in the airgap. Figure 5: Permeance waveform of the modulator.