Keywords

1 Introduction

In the case of artillery, conventional weapons launch projectiles carrying explosive warheads at high velocities to destroy and eliminate enemy targets. Such systems are expensive, dangerous, pose the risk of premature detonation and have a limited range and energy available on the target [1]. To mitigate the above-mentioned problems, the use of electromagnetic launchers capable of launching projectiles at hyper-velocities is being studied as a viable alternative to conventional chemical propellant-driven artillery [2]. Table 1 presents technical superiority of electromagnetic launchers over conventional chemical propellant-driven launchers used in ground combat. The comparison is carried out between a Mauser 30 mm MK-32 Cannon and a traditional electromagnetic launcher.

Table 1 Conventional medium Caliber Cannon versus electromagnetic launcher
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The higher projectile velocity, higher acceleration and higher energy to mass ratio of the electromagnetic launcher in comparison to the Mauser cannon [23] make it a superior alternative to conventional artillery. Electromagnetic launchers especially railguns and coil-guns have been a subject of research since the 1970s, but current research on electromagnetic launcher mainly focussed on the electromechanical parameters of the coil-guns [4, 5]. Along with presenting both computer simulations and hardware implementation of the coil-gun, this paper also speaks on their on-field deployability[5] In the case of electromagnetic launchers, three types of hyper-velocity electromagnetic launchers are being extensively studied viz. railguns, linear induction motor guns and coil-guns [6].

2 Basic Design of Coil Gun

Railguns are beyond the scope of this study, and the main area of focus shall be exclusively on linear induction motor guns and coil-guns. The main operating principle of both these weapons is indicated, i.e. Lenz’s law and the key differences between these weapons are mainly in their mechanical performance which is discussed in the subsequent sections. As shown in Fig. 1, coil-gun is an electromagnetic launcher that consists of a solenoid(s) that acts as the stator and a runner. The principle of operation of a coil-gun is identical to that of a conventional linear induction motor except for the fact that a linear induction motor works on AC whereas coil-gun works on DC.

Fig. 1
figure 1

Schematic of coil gun [2]

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The use of DC in place of AC reduces the effects of the oscillatory nature of the force that is produced because of the alternating nature of the current that is supplied to the stator in the case of a linear induction motor gun. This aspect will be covered in greater detail in the subsequent sections.

3 Modelling of Linear Induction Motor Gun

This section presents a discussion on the modelling of a linear induction motor gun. Figure 2 presents a schematic of a rotary induction motor [1]. The development of a rotary induction motor into a linear induction motor is presented in Fig. 3.

Fig. 2
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Schematic of a rotary induction motor [1]

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Fig. 3
figure 3

Rotary induction motor developed into linear induction motor [1]

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As shown in Fig. 3, a linear induction motor is essentially a conventional rotary induction motor (Fig. 2) that has been cut along an axial plane and then laid on a flat surface [6]. Such a motor consists of a stator which is the stationary part and a runner which is the travelling part.

The stator in a linear induction motor produces a travelling magnetic field instead of a rotating one which is in the case of a rotary induction machine. The runner chases the travelling magnetic field and tries to catch it as per Lenz’s law [6]. The stator and the runner are often referred to as the primary and secondary, respectively, and these terms will be used interchangeably hereafter throughout the paper. The runner shall also be referred to as the projectile. A linear induction motor’s performance can be studied by analysing its equivalent circuit both in the a–b–c frame and stationary frame [6]. The equivalent circuit of linear induction motor in the a–b–c frame is presented in Fig. 4.

Fig. 4
figure 4

Per phase equivalent circuit of linear induction motor in a–b–c reference frame

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The edge effect comes into play in a linear induction motor because the distribution of flux in a linear induction motor is asymmetrical unlike that in a rotary induction motor because of a uniform air gap between the stator and rotor of a rotary induction motor which is absent in a linear induction motor. The end effect factor of the linear induction motor is determined from the expression as follows:

$$Q = \frac{{{\text{DR}}_{{\text{r}}} }}{{\left( {L_{{\text{m}}} + L_{{\text{r}}} } \right)v}}$$
(1)

The function that governs the effect of the edge effect factor on the various properties of the linear induction motor is given by:

$$f\left( Q \right) = \frac{{1 - e^{ - Q} }}{Q}$$
(2)

Now, the correction values of rotor resistance and magnetising induction can be determined with the help of Eq. (2

$$R\prime_{{\text{r}}} = { }R_{{\text{r}}} f\left( Q \right)$$
(3)
$${ }L\prime_{{\text{m}}} = { }L_{{\text{m}}} \left( {1 - f\left( Q \right)} \right)$$
(4)

Now, writing the voltage equation of the linear induction motor in the stationary reference frame, the voltage matrix can be constructed as follow:

$$\left[ {\begin{array}{*{20}c} {u_{{s{\upalpha }}} } \\ {u_{{s{\upbeta }}} } \\ 0 \\ 0 \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {i_{{s{\upalpha }}} } & {i_{{r{\upalpha }}} } & {{\Phi }_{{s{\upalpha }}} } & 0 \\ {i_{{s{\upbeta }}} } & {i_{{r{\upbeta }}} } & {{\Phi }_{{s{\upbeta }}} } & 0 \\ {i_{{r{\upalpha }}} } & {i_{{s{\upalpha }}} } & {{\Phi }_{{r{\upalpha }}} } & {{\Phi }_{r\beta } } \\ {i_{{r{\upbeta }}} } & {i_{{s{\upbeta }}} } & {{\Phi }_{{r{\upbeta }}} } & { - {\Phi }_{r\alpha } } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {R_{{\text{s}}} + R\prime_{{\text{r}}} } \\ {R\prime_{{\text{r}}} } \\ {\frac{{\text{d}}}{{{\text{d}}t}}} \\ {{\upomega }_{r} } \\ \end{array} } \right]$$
(5)

From the above matrix, the equations for flux are determined as follows:

$${\Phi }_{{s{\upalpha }}} = L_{s} i_{{s{\upalpha }}} + L_{{\text{m}}}^{\prime } \left( {i_{{s{\upalpha }}} + i_{{r{\upalpha }}} } \right)$$
(6)
$${\Phi }_{{s{\upbeta }}} = L_{s} i_{{s{\upbeta }}} + L_{{\text{m}}}^{\prime } \left( {i_{{s{\upbeta }}} + i_{{r{\upbeta }}} } \right)$$
(7)
$${\Phi }_{{r{\upalpha }}} = L_{r} i_{{r{\upalpha }}} + L\prime_{{\text{m}}} \left( {i_{{s{\upalpha }}} + i_{{r{\upalpha }}} } \right)$$
(8)
$${\Phi }_{{r{\upbeta }}} = L_{r} i_{{r{\upbeta }}} + L\prime_{{\text{m}}} \left( {i_{{s{\upbeta }}} + i_{{r{\upbeta }}} } \right)$$
(9)

From Eq. (6), the electromagnetic thrust acting on the runner is given by,

$$F_{{\text{e}}} = \frac{{3{\pi p}}}{{8{\uptau }}}\left( {{\Phi }_{{s{\upalpha }}} i_{{s{\upbeta }}} - {\Phi }_{{s{\upbeta }}} i_{{s{\upalpha }}} } \right)$$
(10)

The net mechanical force acting on the block is given by

$$F_{{{\text{net}}}} = m\frac{{{\text{d}}v}}{{{\text{d}}t}} = F_{{\text{e}}} - F_{{{\text{losses}}}}$$
(11)
$$m\frac{{{\text{d}}^{2} x}}{{{\text{d}}t^{2} }} = \frac{{3{\pi p}}}{{8{\uptau }}}\left( {{\Phi }_{{s{\upalpha }}} i_{{s{\upbeta }}} - {\Phi }_{{s{\upbeta }}} i_{{s{\upalpha }}} } \right) - F_{{{\text{losses}}}}$$
(12)

From the above equation, it can be inferred that the force on the runner is oscillatory since the currents in both α and β axes are sinusoidal.

Since, a dedicated block for a linear induction motor is not available on SIMULINK, the above equations were used, and a MATLAB code was written in the MATLAB workspace to study the behaviour of a linear induction motor. The parameters of linear induction motor gun are presented in Table 2.

Table 2 Parameters of linear induction motor gun
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Figure 5 presents the thrust-time graph of a linear induction motor. From Fig. 5, it is clear that the thrust, i.e. the force acting on the runner is highly oscillatory, thereby grossly limiting the destructive force of the gun by limiting the peak velocity due to the fluctuating nature of the direction of the acceleration vector of the projectile. Figure 6 presents the speed-time graph of the runner in a linear induction motor. Figure 6 shows another key limitation of a linear induction motor gun which is the low muzzle velocity of the projectile.

Fig. 5
figure 5

Thrust-time graph of linear induction motor

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Fig. 6
figure 6

Speed-time graph of the runner in a linear induction motor

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4 Performance Analysis of Coil Gun

Figure 7 presents the magnetic field inside a solenoid at any point on the axis of the solenoid. As shown in Fig. 7, each of the coils of a coil-gun can be regarded as a solenoid of finite length with the runner at point P, and its electromagnetic parameter can be determined from Eq. (13).

Fig. 7
figure 7

Magnetic field inside a solenoid

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From Biot-Savart law, the expression for the magnetic field inside the solenoid comes to be:

$$B = \frac{{{\upmu }_{0} i\left( t \right)_{{\text{s}}} }}{2}\left( {\frac{a}{{\left( {x\left( t \right)^{2} + a^{2} } \right)^{0.5} }}} \right)$$
(13)

Force acting on a current-carrying conductor in a magnetic field is given by

$$F = i\left( {\vec{l} \times \vec{B}} \right)$$
(14)

Force acting on the runner is given by,

$$F = i_{{\text{r}}} \left( t \right)\left( {\frac{{l{\upmu }_{0} i_{{\text{s}}} \left( t \right)}}{2}\left( {\frac{a}{{\left( {x\left( t \right)^{2} + a^{2} } \right)^{0.5} }}} \right)} \right)$$
(15)

From Newton’s second law,

$$m\frac{{{\text{d}}^{2} x}}{{{\text{d}}^{2} t}} = i_{{\text{r}}} \left( t \right)\left( {\frac{{l{\upmu }_{0} i_{{\text{s}}} \left( t \right)}}{2}\left( {\frac{a}{{\left( {x\left( t \right)^{2} + a^{2} } \right)^{0.5} }}} \right)} \right)$$
(16)

Solving the above differential equation gives us an estimate of the position of the runner. It must be noted that the above equations have been derived by assuming that the size of the runner is negligible in comparison to that of the stator [11].

5 Simulation Results and Discussion

A coil-gun can be treated as a series R-L load. To meet the extremely high current demand of a coil-gun, it is generally powered by a capacitor bank of high voltage and high capacitance [7, 8] (Table 3).

Table 3 Simulation parameters
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A coil-gun is generally set up as shown in Fig. 8.

Fig. 8
figure 8

Generic block diagram of 3 stage coil-gun

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The capacitors need to be charged up to 400 V DC, and once the capacitors are charged, they are disconnected from the primary power supply [9, 10] and connected across the coils [2, 11]. Which are triggered as per the following sequence:

COIL 1 is turned ON initially when the projectile is initially at rest. The projectile is pulled into the coil. Once fully inside COIL-1, switching circuit 1 is turned OFF and the projectile continues to move through the coil on account of its inertia [12].

The sequential turn ON and turn OFF of each of the coils is necessary for the creation of the travelling magnetic field which will cause the projectile to advance through the barrel and acceleration [13].

Figures 9 and 10 represent the force–time and speed-time graph, respectively, for coil gun. From Fig. 9, it can be observed that the force acting on the projectile although variable in magnitude maintained a positive direction up to 2.71428 ms after reversing its direction for a brief period before becoming zero. This allows the projectile to have a greater peak velocity, i.e. 35 m/s as seen in Fig. 10 for the same supply voltage of 400 V because of the positive direction of the thrust vector for most of its operating period. Most importantly, the projectile fired by a coil-gun reaches its peak velocity within 3 ms, whereas one fired by a linear induction motor gun takes almost 1 s apart from being several orders of magnitude slower. A lower firing time allows for quick reloading of the gun, in this case resulting in a theoretical firing capacity of more around 368.42 shots per second instead of a mere 1 shot per second which could be achieved by the linear induction motor gun. This ensures that a greater number of targets can be engaged with further bolstering the destructive capabilities of the coil-gun w.r.t a linear induction motor gun.

Fig. 9
figure 9

Force–time graph for coil-gun

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Fig. 10
figure 10

Speed-time graph for coil-gun

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6 Conclusion

The increased destructive power, reduced wear and tear and improved efficiency of a coil-gun combined with its reduced operating cost per shot make it a viable alternative to conventional chemical-driven weapons as seen in the introductory section(s) of the paper. Further, in comparison to other electromagnetic weapons, it can be concluded from the above simulations, it can be concluded that coil-guns are superior to linear induction motor gun as the projectile fired by the coil-gun is 35 times faster and can fire 368 shots per second such a high number of shots per second allows a coil-gun to have the range and capacity of an artillery gun but with the firing speed of an infantry weapon both in terms of muzzle velocity and the number of shots fired in each second. The Achilles’ heel for coil-guns is not their range or destructive power but the availability of sufficient electrical power to sustain such a high number of shots per second for the entire course of a given battle if not a war. For coil-guns to become mainstream, advancements in power-supply technology are required so that they can be battle-ready under any circumstances.