# An Electromagnetic Circuit Design to Improve a Multi-Stage Coil-Gun's Energy Conversion Efficiency
## 3 Establishment of the Coil Gun Design Problems  
As shown in Figure 2, the solenoid coil's winding shape is generally determined by the number of turns in its radial (M) and axial directions (N), respectively. Additionally, in this study, we also included the diameter (d) of the copper wire used to manufacture the coil in the coil design variables. In addition to the coil shape, the capacitance and capacitor charging voltage also act as design variables. Furthermore, not only does the coil gun's circuit configuration act as an important design variable but so does the distance (x) between the solenoid coil and the projectile upon launching, i.e., when the coil gun's capacitor discharges. This importance sprouts from the supplied current's variability. Moreover, the magnitude of the electromagnetic force received by the projectile continuously varies according to the latter's location and the passage of time. Hence, to further understand the magnitude of the electromagnetic force generated in the coil, the circuit's current waveform must be calculated. The electromagnetic force generated in the solenoid coil is determined by the coil's number of turns and the used copper wire's diameter. Likewise, the current's waveform flowing in it is determined by the resistance (\(R\)), according to the coil's shape and inductance (\(L\)), and the capacitance (\(C\)). Generally, the coil gun's circuit is composed of a solenoid coil and a capacitor. Due to the coil's self-resistance, the shape of the current flowing in the circuit takes the form of an RLC function and the shape of an underdamped system. Three sub-systems, namely the mechanical, magnetic, and electrical sub-systems, are  
Figure 1: The projectile used in the experiment. Figure 2: The schematic diagram of coil gun. coupled in a coil gun system. This coupled system can be expressed using the following governing equation:  
\[L\frac{d^{2}}{dt}+R\frac{di}{dt}+\frac{1}{C}i\;dt=0 \tag{1}\]  
\[i(0)=0\]  
\[\frac{di}{dt}|_{t=0}=\frac{V_{0}}{L}\]  
\[\mathrm{m}\ddot{x}=F(x)\]  
Additionally, the magnetic force received by the projectile from the coil gun is affected by the magnetic flux density, magnetic permeability, etc. The magnetic flux density continuously varies according to the projectile's position, and the force received by the projectile at position \(x\) is expressed as follows:  
\[F(x)=\frac{du}{dx} \tag{2}\]  
where \(u\) is the magnetic field energy in a certain space, expressed as the product of the volume (\(v\)) and the magnetic flux density (\(B\)). The variation of energy, \(\Delta u\), is expressed as:  
\[\Delta u=v\frac{Bx^{2}}{2}\bigg{(}\frac{1}{\mu_{0}}-\frac{1}{\mu_{m}}\bigg{)} \tag{3}\]  
where \(\mu_{0}\) and \(\mu_{m}\) respectively represent the magnetic permeability of air and the material occupying the space. We can obtain the force F received by the projectile using Equations (2) and (3). We can express the magnetic flux density (\(B\)) as an integral value with respect to the distance as follows, where \(r\) is the projectile's radius and \(L\) is the solenoid coil's length:  
\[F(x)=\frac{du}{dx}=\frac{\pi r^{2}}{2}\bigg{(}\frac{1}{\mu_{0}}-\frac{1}{\mu_ {m}}\bigg{)}\frac{d}{dx}[\int_{x}^{x+L}B(z)^{2}d\zeta\;] \tag{4}\]  
The magnetic field B also changes according to the projectile's distance x. Therefore, simulating the coil gun system expressed above using analytical methods is difficult. Figure 3 is a graph showing the projectile velocity's shape, the electromagnetic force received by it, and the coil gun circuit's current waveform in a system that uses a 7000 uF capacitor on a solenoid coil wound with 24(M) \(\times\) 9(N) turns, using a 1.7 mm diameter copper wire. Upon operating the coil gun, the distance between it and the projectile was 7 mm. Notably, as shown in Figure 3, the projectile received a force in the direction opposing that of its launching while its center of gravity passed through the coil's center at 3.4 ms. Additionally, the received electromagnetic force was proportional to the current's magnitude.