Two fluxes multistage induction coilgun

This paper presents a brand new induction electromagnetic launcher, which uses two magnetic fluxes in order to accelerate a projectile. One magnetic flux induce a current in the armature and the second magnetic flux is creating a radial magnetic field. This aproach offer multiple advantages over single flux designs. First we are able to control the induced current in armature because we use the coil just to induce current inside the ring with a great efficiency. Second advantage is the angle of 900 between magnetic field density B and the ring. We used the induction to avoid contact between armature and accelerator. In order to create the magnetic field radial we used four coils perpendicular on armature. This approach alove us to control the phase difference between induced current in armature and current in magnetic field coils for a maximum force. The phase difference is obtained by changing the frequency of magnetic field coils power source. We used simulation software to analyze, and simulate a multistage induction coilgun design with two fluxes. The simulation results demonstrated the theoretical results.


Introduction
The actual guns to accelerate projectiles based on chemical energy have achieved their limits but the requirements are continuously growing. The electromagnetic launch systems (EMLS) can expand the possibilities of guns [1], [2].
Many EMLS are designed to use the Lorentz force ⃗ ∫ ⃗ ⃗⃗ A well-known design is railgun. The railgun consists of two rails and an armature made by conducting material. The armature is connected with both rails and is accelerated alongside the rails. The armature is accelerated by Lorentz force alongside the rails when the rails are powered by a high current.
In the railgun ( Figure 1) the current I is the same current provided by the power system. The rail creates the magnetic field ⃗⃗ . A high value of the force is obtained when is used a very high value of current I and, therefore, a big power supply system.
Because of this high currents used those electric contacts affect the rails during lunches [3]. In order to reduce the value of current I the rails can be replaced by coils.
The current inside armature is obtained by using induction instead of sliding contacts. (2) where θ is the angle between ⃗⃗ and ⃗⃗ (normal unit of surface area A). We assume the magnetic field is uniform distributed in space.
The coilgun design is based on induction where only the magnetic field is variable, the surface A and angle θ are constant. The Faraday's law can be writhed: One approach consists of a barrel of coils which allow a projectile made by aluminium to move inside them [4]. The magnetic flux created by the coils is the only one magnetic flux, which induces current in projectile and provides the radial magnetic field which interacts with the induced current. It is difficult to control in the same time with one coil the rate of change of the axial magnetic density ⃗⃗⃗⃗⃗ and the radial magnetic flux density ⃗⃗⃗⃗⃗ [5]. In order to increase the radial magnetic flux density ⃗⃗⃗⃗⃗ a design with magnetic circuit made by ferromagnetic materials was proposed ( Figure 2). The E shaped design create a magnetic circuit but does not allow to control the difference of phase between induced current in projectile and the phase of the radial magnetic flux density ⃗⃗⃗⃗⃗ .
Based on this observation we can identify the main aspects which should be taken into consideration when an electromagnetic launch system is designed. First the Lorentz force should be used to accelerate the projectile. Second the contact between the projectile and the accelerator should be avoided. Because of this, coils should be used to induce current into projectile. By using induction the value of current I induced into armature can be higher than the current provided by the power system.
Third the design should give us the possibility to change the phase of induced current in armature and the phase of the radial magnetic flux density ⃗⃗⃗⃗⃗ in order to obtain a high value of the Lorentz force. A coilgun, which obeys all this conditions, is presented in the following section.

A multistage induction coilgun design with two fluxes
First we calculate an air core coil which will induce current into a ring placed outside ( Figure 3). The aluminium ring is placed outside solenoid because it will act as armature, which will be mechanical coupled with projectile.  We need another coil, which will create the radial magnetic field [4]. Usually the armature in a coilgun is placed inside the coil and the current I and magnetic field B are produced by the same coil. For our design the magnetic field B is produced by others air core coils placed around ring at θ=90 0 in order to obtain maximum force on the ring ( Figure 4). The simplified drawing of this circuit is displaied in Figure 6. We chose to represent only the induction coil Lb, the ring Li and the coil which produce the magnetic field Lc. The ring is a short circuit but the coils are powered by two sinusoidal sources with following expression.
( ) (6) ( ) (7) Because all coils are identical we have the following equality: and Applying the KVL we obtain the following system of differential equations: The expressions of currents are: Because the time when the ring is accelerated on a stage is very short (below 10 ms) the Lorentz force acting on ring depend only by the first semi period current oscillation at 50 Hz.
The expression of currents can be approximated: The magnetic field created by the L c solenoid can be calculated according with Ampere's law: In our case i enclosed =i c and the expression of B at the end of solenoid L c became: where: N=number of turns of solenoid (dimensionless); r = radius of coil (meters); h = length of solenoid (meters). The expression of Lorentz force became: In order to find the maximum value of Lorentz force F we should find the maximum of expression: Because the above expression does not have a global maximum we maximize each expression of the sum. We obtain the following situation: First situation is obtained when but it is very difficult to obtain the same phase for induced current and current inside magnetic field coils because of mutual induction between coils. The expression of force becomes: Because we approximated the expression of currents for ideal conditions it is not possible to obtain the same values for phase but we are looking for a maximum value of force between those values.
The following dimensions of this design were calculated [5]. The mass of aluminium ring was calculated at 10 grams. We started from dimensions of device used for E shaped. The Maxwell interactive software package that uses the finite element method (FEM) was used to analyse, and simulate two fluxes multistage induction coilgun design [7], [8].

The results of simulation
The coilgun with these dimensions was created using 3D simulation software. The values of voltages and duration of pulses for every stage are presented in Table 2. In this simulation all coils from the same stage were powered with the same voltage.  frequency of power source of magnetic field coils was changed to 120 Hz with the same phase zero. The resistance of coil was decreased at a very low value in order to obtain the phase 90 0 for current. The continuous line with triangle is the time variation of the current inside the magnetic field coil. At 6.6 ms when the ring left the stage 1 the power line is switched off and the value of currents inside stage 1 coils drop rapidly to zero value.  The time variation of current induced into the ring is represented by long dash. The maximum value of induced in ring current is 13,973.6 A when the ring is in first stage and 40,236.2 A when the ring is in second stage. The maximum value of current used to induce the current into the ring is 2,332.9 A inside the induction coil from stage 1. All other coils are working with currents below this value. From this point of view, our design is better than the railgun because the value of current inside armature is higher than power supply current. The flux linkage created by stage 1 is presented using a continuous line with circle and for stage 2 is a continuous line with triangles.  The time variation of speed of ring is presented with continuous line and with long dash is presented the position of ring during acceleration time. The ring left stage 1 at time 6.49 ms with speed 64.94 m/s and the stage 2 at time 7.23 ms with speed 101.96 m/s. The maximum speed 106.98 m/s is achieved at time 7.33 ms. After that time the ring is no longer accelerated and the speed drop rapidly to zero because the simulation length is 150 mm.

Conclusion
The multistage induction coilgun with two fluxes is better than single flux designs. The induced current in armature can be controlled much efficient because we use the coil just to induce current inside the ring with great efficiency. The magnetic field density B inside the air gap is perpendicular on ring and the Lorentz force accelerate the projectile during entire length of a stage.
We used the induction to avoid contact between armature and accelerator and to obtain a higher value of induced current than power current.
In order to create the radial magnetic field we used four coils perpendicular on armature. This approach give us the possibility to control the phase difference between induced current in armature and current in magnetic field coils in order to obtain the maximum force. The phase difference is obtained by changing the frequency of magnetic field coils power source. We do not need less electrical energy because is expensive but because we can use a small power source.