## 3 Simulation Results and Analysis
### Mechanism Analysis of the Interaction between the Previous and Next Drive Coils and the Armuture  
The presence of induced currents in the drive coils can have an impact on the acceleration effect of the armature. Therefore, an analysis and study on this effect is conducted. The electromagnetic force exerted on the armature in the drive coils can be expressed as:  
\[F=\frac{\mathrm{d}M}{\mathrm{d}x}i_{d}i_{p} \tag{11}\]  
In the equations, \"\(i_{d}\)\" represents the current in the drive coil, \"\(i_{p}\)\" represents the induced current in the armature, and \"\(dM/dx\)\" represents the gradient of mutual inductance with respect to the position of the armature. The electromagnetic force exerted on the armature by the preceding-stage drive coil can be expressed as:  
\[F_{n-1}=\frac{\mathrm{d}M}{\mathrm{d}x}i_{d(n-1)}i_{p} \tag{12}\]  
The electromagnetic force exerted on the armature by the subsequent-stage drive coil can be expressed as:  
\[F_{n+1}=\frac{\mathrm{d}M}{\mathrm{d}x}i_{d(n+1)}i_{p} \tag{13}\]  
Figure 11: Axial force density (with units of \(\mathrm{kN/m^{3}}\)) of the armature with an inter-stage distance of \(D=15\) mm: (**a**) PP at 1.62 ms; (**b**) PP at 2.38 ms; (**c**) PP at 2.97 ms; (**d**) PN at 1.74 ms; (**e**) PN at 2.54 ms; and (**f**) PN at 3.13 ms.  
As can be seen from Equations (11)-(13), the electromagnetic force exerted on the armature is proportional to the gradient of mutual inductance between the armature and drive coil, as well as the currents in both the armature and drive coil. When the armature is in front of the center of the drive coil, the mutual inductance gradient is positive. If the currents in the armature and drive coil flow in the same direction, the electromagnetic force is a driving force; if they flow in opposite directions, the electromagnetic force is a braking force. When the armature passes through the center of the drive coil, the mutual inductance gradient becomes negative. If the currents in the armature and drive coil flow in the same direction, the electromagnetic force is a braking force; if they flow in opposite directions, the electromagnetic force is a driving force. Figure 12 displays the current density distribution at different times. In Figure 12a, at 0.4 ms, the induced current in the armature is in the opposite direction to the current in the first-stage coil, resulting in a driving force on the armature. In Figure 12b, at 0.8 ms, the induced current in the armature is in the same direction as the current in the first-stage coil, resulting in a braking force on the armature. This article analyzes the entire process of the armature being accelerated by the corresponding drive coil, dividing it into an acceleration section and a deceleration section influenced by the armature capture effect, as shown in Figure 13. In the PN configuration coil launcher, the induced current in the previous and next drive coils is opposite to the current flowing in the nth-stage coil. As can be seen from Equations (1)-(3), during the acceleration stage of the nth-stage coil, the (n \(-\) 1)th-stage coil provides acceleration force to the armature, while the (n \(+\) 1)th-stage coil provides braking force to the armature. During the deceleration stage of the nth-stage coil, the (n \(-\) 1)th-stage coil provides braking force to the armature, while the (n \(+\) 1)th-stage coil provides acceleration force to the armature. In the PN configuration coil launcher, the induced current in the (n \(-\) 1)th-stage coil is in the same direction as the current flowing in the nth-stage coil, and there is no induced current in the subsequent stages when the nth-stage coil is energized. During the acceleration stage of the nth-stage coil, the (n \(-\) 1)th-stage coil provides braking force to the armature, while during the deceleration stage of the nth-stage coil, the (n \(-\) 1)th-stage coil provides acceleration force to the armature.