## 3 Variable Pole Pitch (VPP) Electromagnetic (EM) Propulsion
### Numerical Analysis of VPP EM Propulsion  
A flowchart to numerically analyze the kinetics of the VPP EM propulsion is presented in Figure 4. The numerical calculations in Figure 4 have been realized in MATLAB, including the design of the LS-secondary DLIM for each section and the analysis of their characteristics. Figure 3: Schematic diagram showing the operation of variable pole pitch (VPP) electromagnetic (EM) propulsion. As part of the numerical analysis, we designed and calculated four configurations of VPP EM propulsion, which were classified into the four categories listed in Table 1. The analysis procedure followed that depicted in the flowchart in Figure 4. The VPP EM propulsion configurations were segmented into three sections, where Section 1 is the head while Section 3 is at the terminal end. The primary slots per pole per phase equaled to 1, 2, and 3, respectively. The VPP EM propulsion system with a supply voltage of 10 kV was designed to meet the requirements of the EM launching system at \(V_{\mathrm{T}}\) = 0.8 Ma, with a loading mass of 24 t, while the system with a supply voltage of 220 V was intended for use in a small-scale platform. In practice, the small-scale DLIM can be used for applications in industrial mechanics that have restrictions in terms of the maintenance expense and supply frequency, which requires more limitation on both the maintenance expense and supply frequency, because back-and-forth translation tends to cause more ohmic losses in the core and higher risk of mechanical cutting in these occasions. According to the theory of electrical machines, the thrust in each section is correlated with the secondary speed. Thus we can derive the kinetic characteristics, including the displacement and performance period, as follows:  
\[v=(1-s)V_{s},\ F_{e}(s)=m\frac{dv}{dt},\ S=\int vdt,\ T=\int 1 dt \tag{15}\]  
Then a substitution in Equation (16) can be carried out in integration of the kinetic characteristics:  
\[dt=-\frac{mV_{s}}{F_{e}(s)}ds \tag{16}\]  
Therefore, the displacement and performance period of VPP EM propulsion yield in Equation (17):  
\[S=m\underset{i}{\sum}\left[V_{si}^{2}\underset{s_{0i}}{\int}\frac{1-s}{F_{ei} (s)}ds\right],\ T=m\underset{i}{\sum}\left[V_{si}\underset{s_{0i}}{\int} \frac{1}{F_{ei}(s)}ds\right],\ s_{0i}=1-\frac{V_{s,i-1}}{V_{si}}, \tag{17}\]  
where the subscript \(i\) represents the parameters belonging to the \(i\)th section. The kinetic characteristics in Table 1 were calculated using Equation (16). In contrast to VPP EM propulsion, CPP EM propulsion adopts primaries in Section 3. The displacement of the CPP EM propulsion system with a supply voltage of 10 kV, and that for the one with a supply voltage of 220 V were 6064.36 m and 69.77 m, respectively, while the performance  
Figure 4: Design and calculation flowchart of the VPP EM propulsion. periods were 42.13 s and 6.32 s, respectively. By comparing the two types of EM propulsion, we concluded that VPP EM propulsion has better performance in large scale for both the supply frequency and kinetic characteristics. In contrast, the small-scale propulsion system exhibited mixed results when the VPP EM propulsion method was adopted, because the VPP EM propulsion method had makes less desirable kinetics, even though it led to an optimal improvement in the supply frequency, which ultimately results in lower ohmic losses. The EM shear stress in Table 1 for each DLIM has been calculated via Equation (11). Since the DLIM usually operates with an EM shear stress of 70-100 kN/m\({}^{2}\)[17], the machines in Table 1 are considered practical. The structural parameters can be obtained from Equation (12). Owing to the principles of VPP EM propulsion and restrictions in Equations (13) and (14), the maximum structural parameters for the large-scale VPP EM propulsion will occur at Section 1 with variable \(V_{\mathrm{s}}\). Thus, the maximum values turn out to be: \(\sigma_{\mathrm{max}}=75.0\) MPa, \(\tau_{\mathrm{max}}=2.76\) MPa and \(k_{\mathrm{max}}=4.59\) mm.