An experimental schematic for a railgun is shown in Fig. 7.41. With switch Sw-2 open, switch Sw-1 is closed and the power supply charges the capacitor bank to 10 kV. Then switch Sw-1 is opened. The railgun is fired by closing switch Sw-2. When the capacitor discharges, the current causes the foil at the end of the gun to explode, creating a hot plasma that accelerates down the tube. The voltage drop in vaporizing the foil is negligible, and, therefore, more than 95% of the energy remains available for accelerating the plasma. The current flow establishes a magnetic field, and the force on the plasma caused by the magnetic field, which is proportional to the square of the current at any instant of time, accelerates the plasma. A higher initial voltage will result in more acceleration.
The circuit diagram for the discharge circuit is shown in Fig. 7.42. The resistance of the bus (a heavy conductor) includes the resistance of the switch. The resistance of the foil and resultant plasma is negligible; therefore, the current flowing between the upper and lower conductors is dependent on the remaining circuit components in the closed path, as specified in Fig. 7.41.
The differential equation for the natural response of the current is
\frac{d^{2}i(t)}{dt^{2}} + \frac{R_{bus}}{L_{bus}}\frac{di(t)}{dt} + \frac{i(t)}{L_{bus}C} = 0
Let us use the characteristic equation to describe the current waveform.
