## 1 Introduction  
In recent times, electromagnetic launching systems have been adopted to substitute existing launch systems. For example, electromagnetic launchers (EMLs) or railguns have been developed to install battleships by the US Navy and the Chinese Navy since 1995 and 2011, respectively [1; 2; 3]. Moreover, the French-German Research Institute has developed the PEGASUS Launcher, having a 10-Mega Joule (M)) pulse power system, to increase efficiency and high muzzle velocity [4; 5; 6]. The most attractive advantage of an EML is the ability to obtain higher projectile velocities than conventional gunpowder-type guns. Another application of an electromagnetic launching system is the electromagnetic aircraft launch system (EMALS). This is currently being developed by General Atomics and the US Navy. The main advantage of this is that it can accelerate an aircraft more smoothly, thereby incurring less stress on the aircraft and carrier. Further, EMALS has less weight, cost, and maintenance requirements than a steam catapult [7]. Similar to EMALS, the magnetic system for UAV carriers has been developed. The advantages of this system include a short runway, reduced risk of pilot casualty, and reduced cost of having an airborne device [8]. Another application is that of a heavy projectile launcher. The direct launch of payloads into orbit is impossible. However, it would be possible to launch rugged payloads, such as fuel, water, and others. There are two main advantage of this technology: it is possible to achieve high muzzle velocities and it is easy to force control with a pulse power system [9; 10; 11; 12]. Furthermore, the propulsion force of an electromagnetic launcher is the Lorentz force; this force is applied electric propulsion (EP). EP is termed ion or plasma thrusters. It has a low thrust level compared to that of electromagnetic launchers. EP has significant advantages for in space propulsion for smaller objects and longer durations. EP is independent of a propellant, posing a risk of explosion with a large energy density [13; 14; 15; 16]. Since the 1950s, studies and research on EMLs have been conducted and a few useful equations to predict their performance have been established. A useful equation is given below, which is schematically illustrated in Figure 1.  
\[F=\frac{L^{\prime}I^{2}}{2} \tag{1}\]  
This equation draws from the conservation of energy requirement  
\[W_{delivered}=W_{mech}+W_{ind} \tag{2}\]  
where \(W_{mech}\) is calculated by \(F_{Prop}\) generated on a projectile of mass \(m\) and displacement (\(dx\)) from state 1 to state 2. \[W_{mech}=\left(F_{Prop}\right)dx \tag{3}\]  
Further, the increase in inductive energy \(W_{ind}\) is calculated using inductance gradient (\(L^{\prime}\)) determined from \"Kerrisk \(L^{\prime}\),\" which is a function of the shape of the rail and current (\(I\)) [17; 18]. \[W_{ind}=\frac{dLI^{2}}{2}=\frac{L^{\prime}dxI^{2}}{2} \tag{4}\]  
The energy delivered to the EML circuit \(W_{delivered}\) is  
\[W_{delivered}=Vldt=L^{\prime}I^{2}xdt=L^{\prime}I^{2}dx \tag{5}\]  
The voltage form of [PERSON]'s law is given as \(V=\frac{d\Phi}{dt}\). The voltage is then given by  
\[V=\frac{d(LI)}{dt}=\frac{L^{\prime}dxI}{dt}=L^{\prime}Iv \tag{6}\]  
Figure 1: Electromagnetic launcher (EML) schematic view. Equations (2)-(6) indicate that the propulsion force of an EML (Equation (1)) generated on the projectile can be calculated from the inductance gradient and current at a specific time [19]. Using this equation, considerable analytical research was conducted to predict the performance of an electromagnetic launcher or other [20; 21; 22; 23; 24]. However, these analytical analyses did not consider variations in specific time current with an increase in resistance and inductance. Moreover, this equation applies to active electromagnetic launcher armor that utilizes the Lorentz force in a manner similar to an electromagnetic launcher [25]. To identify the propulsion Lorentz force at a time instant, current (\(I\)) should then be determined.