## 3 Simulation Results and Analysis
### Results and Verification  
The research conclusion can be summarized by the following equations. Where \(L_{d}\) represents the length of the driving coil, \(D\) represents the distance between driving coil stages, and \(T\) represents the threshold of the distance between driving coil stages, which is obtained by the intersection of the two curves in Figure 14.  
As shown in Equation (14), in this study, \(K\) is defined as the ratio of the threshold \(T\) to the length \(L_{d}\) of the driving coil. \[K=\frac{D}{L_{d}}\times 100\% \tag{14}\]  
When \(D=T\), there is a theoretical threshold \(T\) for the distance between the driving coil stages. At this point, when the launch device emits the armature by applying power in either the PP or PN methods, although there are differences in the acceleration principles, the same acceleration effect will be achieved. Let \(K\) at this time be denoted as \(K_{T}\) and \(K_{T}\) can be calculated as follows:  
\[K_{T}=\frac{T}{L_{d}}\times 100\%=\frac{7.5\;\text{mm}}{20\;\text{mm}}\times 1 00\%=37.5\% \tag{15}\]  
When \(K<K_{T}(D<T)\), the driving coils of the launch device are powered by the PP method. The acceleration effect on the armature will gradually decrease with an increase in inter-stage distance. If powered by the PN method, the acceleration effect on the armature will also gradually decrease with an increase in inter-stage distance. However, the PP method is always superior to the PN method. Therefore, in this case, it is recommended to use the PP method for the launch device. When \(K>K_{T}(D>T)\), the driving coils of the launch device are powered by the PP method. The acceleration effect on the armature will still gradually decrease with an increase in inter-stage distance. If powered by the PN method, the acceleration effect on the armature will still gradually decrease with an increase in inter-stage distance. However, the PN method is always superior to the PP method. Therefore, in this case, it is recommended to use the PN method for the launch device. From Equations (14) and (15), it is evident that the optimal power-on method of the launch device is determined by the distance \(D\) between different stages based on the parameters \(T\) and \(K_{T}\) of the device. The \(T\) values in each launch device vary due to the device's different size specifications, while the ratio \(K\) is not influenced by the device size. In this study, it is approximated that \(K_{T}=37.5\%\). In order to validate and confirm the research findings, ensuring the accuracy and reliability of the study, the analysis results of this research will be applied to the research  
Figure 14: The optimal exit velocity of the armature at different inter-stage distances. device model in the existing literature, verifying the consistency with the results of this study under different models of coil launch devices. In reference [12], where \(L_{d}=16.3\) mm and \(D=10\) mm, the value of \(K\) can be calculated using the following equation:  
\[K=\frac{D}{L_{d}}\times 100\%=\frac{10\text{ mm}}{16.3\text{ mm}}\times 100\%=61.3\%>K_{T} \tag{16}\]  
The research findings that \"The launch velocity in the PN launcher is more than in the PP launcher\" are consistent with the optimal power-on method selection in this study. In reference [13], the launch device adopts the PN power-on method. Simulation is conducted with \(D\) ranging from 5 mm to 40 mm. The research shows that as the value of \(D\) increases, the efficiency of the device's energy conversion gradually improves, which is consistent with the conclusion of this study. In reference [18], where \(L_{d}=100\) mm and \(D=20\) mm, the value of \(K\) can be calculated using the following equation:  
\[K=\frac{D}{L_{d}}\times 100\%=\frac{20\text{ mm}}{100\text{ mm}}\times 100\%=20\%<K_{T} \tag{17}\]  
The research finding that \"In the design process of a multi-stage synchronous induction coil launcher, adjacent driving coils should be arranged with the same polarity\" is consistent with the optimal power-on method selection in this study.