*2.1. Physical Concept*

Magnetic field in a looped circuit composed of magnetic material and air gap tends to be maximized when a current is applied. Hopkinson law used in magnetic circuits tells that: *N I* = *R*Φ, with:


Increasing the magnetic field is similar to an increase of Φ. For a given current and number of coil turns, this increase can be done by reducing the reluctance *R* of the magnetic circuit. Its expression is

$$R = \frac{1}{\mu\_0 \mu\_r} \frac{l}{S}, \text{ with:}$$


The way to reduce reluctance is to minimize the air gap in the magnetic circuit, replacing it by a portion of the iron plunger (having a relative magnetic permeability equal to 5000 or more). Thus, under a strong coil current, the plunger will be propelled in order to reduce the air gap, with an important force dependent on the coil current and the number of coil turns. Figure 8 shows a three-stage coil gun. In this case, coil 1 is powered first, then coil 2, then coil 3. This is the principle of a multi-stage variable reluctance actuator.

**Figure 8.** Three Coils Electromagnetic Launcher Principle [14].

Our case study focuses on coil gun implementations having one, two, three or four coils, with a fixed overall size and quantity of copper, as shown in Figure 9. It is important to note that between each coil, an iron plate has been placed in order to close the magnetic circuit around each coil.

**Figure 9.** Coil gun configurations with 1, 2, 3 and 4 coils sharing the same quantity of copper.

In our case study, each coil is powered by an identical capacitor. These capacitors have an global overall capacitance equal to (4700 μF). This capacitance is split into *n* smaller equal ones, where *n* is the number of coils. Each capacitor can be discharged, one at a time, in its corresponding coil producing a strong current which generates a magnetic force. The iron rod, mobile part of the magnetic circuit slides in a stainless steel tube in order to reduce the air gap of the magnetic circuit. This iron rod is attracted and accelerated as long as the air gap can be minimized. It is slowed down if the plunger goes to far and the air gap increases again. To avoid that, the duration of the current pulse in each coil has to be limited in time.

Discharge from the capacitor to the coil inductor can be described by a second order *RLC* differential equation. This equation has non-constant coefficients because the value of the inductor highly depends on the value of the current in the coil and on the plunger position in the sliding tube. This mixed non-linear model combining electrical and mechanical inputs will be presented in the following sections.
