*2.3. Electrical Model*

At each stage of the coil gun, a <sup>4700</sup> *<sup>n</sup>* <sup>μ</sup>F, 450 V capacitor *<sup>C</sup>* is discharged in the inductance *<sup>L</sup>* using a controlled switch based on a MOSFET Transistor as shown in Figure 13. In this *LC* circuit, resistor *R* must be considered because its value is not negligible at all due to the important number of loops in the coil. *<sup>R</sup>* <sup>=</sup> *<sup>ρ</sup><sup>L</sup> <sup>S</sup>* can be evaluated or measured, where *<sup>ρ</sup>* is the resistivity of copper, *<sup>L</sup>* the total length of the coil wire and *S* the surface of a wire section. This leads to the differential Equation (1) where *L* is not constant, but depends on the plunger position and on the coil current. Considering that, Equation (1) must be solved by numerical simulation.

$$\frac{d^2\mathcal{U}\_\mathbb{C}}{dt^2} + \frac{R}{L}\frac{d\mathcal{U}\_\mathbb{C}}{dt} + \frac{1}{LC}\mathcal{U}\_\mathbb{C} = 0\tag{1}$$

**Figure 13.** Electric circuit.

As shown in Figure 14, *FEMM 4.2* simulations shows that *L* inductance varies by a factor 20 from *L* = 13 mH to *L* = 253 mH in a 1 coil kicking system.

**Figure 14.** Variation of the inductance value depending on the plunger position and the coil current.

The inductance value depends on the plunger position and the saturation of the magnetic circuit due to the current in the coils. In Figure 14, discrete values of the inductance are calculated for positions of the plunger (Figure 15) varying from *x* = −100 mm to 60 mm by increment of 5 mm, and for coil currents varying from 1 A to 200 A by increment of 50 A.

It is important to note that:


**Figure 15.** RoboCup reluctance coil gun kicking system.
