# An Electromagnetic Circuit Design to Improve a Multi-Stage Coil-Gun's Energy Conversion Efficiency
## 4 Finite Element Analysis  
We conducted the coil gun model's finite element analysis using ANSYS Maxwell, a commercial electromagnetic analysis FEA software. Particularly, three coil gun systems using a copper wire diameter of 1.3, 1.7, and 2.0 mm, respectively, were analyzed and compared. Their respective analysis models were configured according to the design variables shown in Table 1, and the maximum projectile velocity of each was derived. Additionally, to shorten the analysis time, we only analyzed the respective 2D models of the coil guns. Figure 4 shows the coil gun's 2D finite element analysis model. In the model, the design variables were set such that the RLC function's period was sufficiently long for the projectile to receive at least 70% of the electrical energy while moving in the launch direction. Moreover, the values of the design variables at which the optimal projec \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Copper Wire** **Diameter (mm)** & **Number of Turns in the Axial Direction, N** & **Number of Turns in the Radial Direction, M** & **Capacitance (uF)** \\ \hline
1.3 & 24–27 & 9–12 & 3000–6000 \\ \hline
1.7 & 23–26 & 9–12 & 5000–8000 \\ \hline
2.0 & 22–25 & 9–12 & 7000–10,000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The ranges of design variables for single-stage coil guns by copper wire diameter. Figure 3: The projectile velocity, electromagnetic force, and supplied current evolutions. tile velocity was achieved were set as reference values. Notably, coil guns that accelerated at a stopped projectile were simulated and compared based on the final projectile velocity when the projectile was completely ejected out of the solenoid coil's magnetic field. The optimal values of the coil gun design elements were derived using the experiment method's design, which is the most efficient method because the design elements of the coil gun are independent of each other. Table 1 shows the ranges of design variables according to the diameters of individual copper wires. A table of orthogonal arrays was created with the set ranges of design variables to compose the test points, and the design variables with the highest velocity were summarized. Table 2 shows the optimal coil gun shapes according to respective copper wire diameters obtained from the finite element analysis. Furthermore, the respective energy conversion efficiencies of the coil guns were compared using simple equations. Assuming that there was no loss during conversions, the energy conversion efficiency was calculated by comparing the electric energy stored in the capacitor and the projectile's kinetic energy. In electric energy, C is the capacitance and V is the charging voltage. In kinetic energy, \(m\) and \(v\) are the projectile's mass and velocity, respectively. \[\text{E}_{\text{electric}}=\text{CV}^{2}\]  
\[\text{E}_{\text{kinetic}}=\frac{1}{2}mv^{2}\]  
For the chosen values of the coil gun variables in the analysis, between the coil gun systems using 1.3 and 1.7 mm diameter copper wires, respectively, the kinetic energy of the projectile was 9% higher. However, the used electrical energy for energy conversion was 133% higher for the system using a 1.7 mm diameter copper wire. Generally, for high energy conversion efficiency, it is good to use a copper wire with a small diameter to \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline
**Copper Wire** **Diameter (mm)** & **Number of Turns in the Axial Direction, N** & **Number of Turns in the Radial Direction, M** & **Capacitance (uF)** & **Projectile Velocity (m/s)** \\ \hline
1.3 & 26 & 9 & 3000 & 33.7 \\ \hline
1.7 & 24 & 9 & 7000 & 35.2 \\ \hline
2.0 & 22 & 10 & 10,000 & 30.4 \\ \hline \end{tabular}
\end{table} 
Table 2: Results of the finite element analysis giving optimal coil gun shapes according to the copper wire diameters. Figure 4: The axisymmetric 2D model of coil gun in MAXWELL.