Effect of driving current profile on acceleration efficiency of electromagnetic railgun

“Railgun,” being a simple electromagnetic launching device, finds utility in various fields ranging from plasma thruster,¹ fuel pallet injector,² space debris destructor,³ to ballistic study of the aerospace machineries.⁴ Although the working principle of the system is very simple, the kinematic behavior of the projectile primarily depends on various design parameters and circuit topologies that decide the governing forces responsible for the projectile acceleration. A comparative estimation of projectile velocity that may be achieved using a railgun launcher in response to sinusoidal under-damped and unidirectional over-damped input current profiles has been reported in this paper. A mathematical model integrating dynamic resistances, frictional effects, and the aforementioned driving current profiles has been developed for investigating and analyzing the overall performance. The obtained simulation results have been validated with the experimentally measured velocity of projectiles in our indigenously developed electromagnetic launcher at the “RAFTAR” facility.⁵ 

The mathematical model mainly comprises primary circuit parameters, such as inductance, resistance, and capacitance. These circuit parameters predominantly decide the amplitude of peak current, pulse length, and current profile. With the instantaneous current, the instantaneous governing force can be calculated, which undergoes integration with a time step of 1 µs to get the desired kinematic parameters, such as velocity and displacement.

The electromagnetic governing force is modified in each step with the implementation of frictional forces, which depends upon velocity and current. In addition, with gradual movement of the projectile, resistance and inductance vary dynamically, and it is updated to the current profile equation at each time step. The schematic of the implemented algorithm is shown in Fig. 1.

As shown in the schematic in Fig. 2, a railgun launcher mainly comprises three blocks, i.e., (i) pulsed power system, (ii) pulse shaping unit, and (iii) load, i.e., “rail–armature” assembly. The pulsed power system constitutes the module “C,” i.e., a capacitor bank made from 16 parallel connected 178 µF, 15 kV energy storage capacitors along with a constant current power supply (CCPS) for charging. For the protection of CCPS, a resistance “R” and a diode “D” are used in series to prevent any damage due to current reversal during discharge.⁵ A pneumatically operated switch “SW” is used for decoupling the CCPS during discharging of the capacitor bank. For pulse shaping, a 10 µH inductor “L” was used to increase the pulse length of the capacitor discharge, which was preceded by ignitron switches “IG₁” and “IG₂.” Triggering IG₁ causes discharge of the capacitor module through the inductor and subsequently to the “load,” and switching of IG₂ crowbars the module. IG₂ has been triggered when the input current pulse reaches its first peak (T/4 of the waveform), which is 200 µs for our present circuit. The short circuit current flowing through the crowbarred channel for an input peak current of 220 is 200 kA.⁵ The solid armature placed in between the parallel rails is the dynamic load.

The parameters governing the drag force are “A_p,” i.e., the area of the front end of the armature; “C_d,” i.e., the drag coefficient with a value of around 0.1; and “ρ_air,” i.e., the density of air.⁹ Due to the small dimensions of the projectile used, retardation due to air drag is very less in our experimental setup as far as short-range investigations are concerned.

The railgun’s electrical circuit parameters that have been considered in our simulation are shown in Table I. In the case when a pulse shaping inductor has been used and a crowbar switch was triggered at the first quarter time period of the injected current in the circuit (i.e., at about delay ∼200 µs from the start of current flowing in the bitter-coil inductor), the cumulative equivalent series inductance and resistance of the circuit were 11.4 µH and 7.5 mΩ, respectively.

The simulated current profile shown in Fig. 3 is found to be well in agreement with the experimentally measured current trace obtained at 12 kV discharge.⁵ 

The simulated waveforms of current, velocity, and displacement of the armature projectile weighing ∼8 g is collectively shown in Fig. 4.

From the results shown in Fig. 4, it is evident that due to friction, the armature does not start moving instantaneously after the arrival of the current pulse; rather, it takes ∼168 µs to overcome the frictional force and then its displacement starts. The unipolar driving current pulse leads to consistent increment in the velocity and displacement of the armature. The temporal variation of the dynamic resistance components is shown in Fig. 5.

From the plot, it is clear that the resistance of rails, i.e., “R_g,” is the most dominant resistive component. The results shown in Fig. 5 infer that with the movement of the projectile, all the dynamic resistance components start increasing. The velocity for this configuration was found to be 931 m/s with projectile weighing ∼8 g. At the exit time, the total dynamic resistance was found to be 0.90 mΩ, while “R_g” was 0.57 mΩ and “R_backemf” was 0.34 mΩ (i.e., back-EMF resistance). This corroborates that “R_g” contributes ∼63% of the total dynamic resistance.

Our next investigation was to simulate the kinematic behavior of the armature when the current is directly discharged from the capacitor bank without any intermediate pulse shaping circuitry. In this case, the pulsed discharge current profile is an under-damped sinusoid. The experimental and simulated discharge current profiles for a peak current of 290 kA are collectively shown in Fig. 6.

The ratio of ac to dc resistance shows variation depending on the width-to-breadth ratio of the rails because the slope “m” and the intercept “C” possess different values for different width-to-breadth ratios of the rails. Forbes and Gorman¹¹ presented a comprehensive experimental analysis and presented a graph depicting “r_ac/dc” vs “p” for varying width to breadth ratios and frequencies. In our setup, the width-to-breadth ratio of the rails is 1.56. From their work, we have also observed that “r_ac/dc” vs “p” graph for width to breadth ratios 1:1 and 2:1 are parallel and it was found to be “0.4.” So, this slope “m” value has been considered for width-to-breadth ratios ranging from 1:1 to 2:1.¹¹ 

It may be noted from the results shown in Fig. 7 that implementation of a new resistive model (model-2) leads to a close match with the experimentally obtained discharge current profile compared to the previous dynamic resistance model, which was used in crowbar topology earlier. Model-2 possess more implicit dependencies to the electrical parameters than model-1, which is why it could provide better fit to the experimental current profile. The resistive model-2, being a function of frequency “ω (t)” of the under-damped sinusoidal current profile, can take into consideration the effect of the dynamic variation of the electrical parameters occurring due to the propulsion of the projectile. Although model-1 also incorporates displacement and temporal dependencies, this model may be too simple to be fit to predict the oscillatory current fed. The initial inductance and resistance of the circuit of ∼4.2 µH and ∼3.5 mΩ also substantiate the quarter time period of the measured discharge current trace. The typical kinematic profile of the accelerated armature projectile of mass 9.73 g along with an under-damped sinusoidal discharge current trace predicated by “model-2” is collectively shown in Fig. 8.

The results shown in Fig. 8 clearly indicate that as a consequence of the oscillatory damped sinusoidal driving current, a subsequent increase in armature velocity is periodic in each half cycle. It increases while the oscillating current increases and then retards for the duration when current decreases. Thus, the oscillatory current profile results in the inconsistent acceleration of armature and reduced velocity for armature projectile. Experimental evidences that a confirm significant reduction in efficient acceleration of armature with a damped sinusoidal driving current input are as follows: (i) for a projectile mass of 9.73 g with a damped sinusoidal current input of 290 kA, a velocity of 675 m/s was achieved experimentally and our simulation model estimates 666 m/s; (ii) on the other hand, for a projectile mass of 10.6 g with a unidirectional current input of 200 kA, the experimentally measured velocity was 689 m/s, whereas the simulation model estimates 612 m/s. In the both the aforementioned cases, the obtained simulation results from the conceived model are well in agreement with experimental results. Furthermore, using this model, the exit velocity of an 8 g projectile for a peak current of 220 kA was compared for both unidirectional and sinusoidal discharges. Through the simulation, it was observed that the armature projectile velocity decreased to 489 from 931 m/s when a damped sinusoidal current was supplied to the rails instead of a unidirectional current. All experimental and simulated results are comprehensively summarized in Table II.

The marginal underestimation of the simulated velocity (in the typical range of 5%–10%) may be attributed to (i) the slight overestimation of the frictional force generated due to the current itself and (ii) also in actual experiments, the minor mass loss that does occur in the armature during acceleration inside the barrel due to the partial melting of contact surface as a consequence of excessive heating and friction with adjacent rails.

The difference between the simulated and experimental velocity estimations suggested that our mathematical model can perform better for an over-damped unidirectional current input than for a damped sinusoidal current. This is because in case of unidirectional current, the overall velocity achieved by the projectile is more than that usually achieved by the sinusoidal one. This leads to more dominance of the effect of the dynamic components of the electrical parameters than the static one. Predicting the exact behavior at higher velocities is difficult for any empirical model and may incorporate errors while estimation.

It has been observed that for both the topologies, a delay of ∼150–200 ms has been taken up by the projectiles for starting displacement after the current pulse is injected. Although the complete nullification of this delay may not be possible due to frictional effects of the armature that arise due to electromagnetic reasons, some minimization may be possible: (i) first, by multi-step firing: one solution may be firing a lower peaked current first, and then when the projectile starts moving, we may inject more current. This will lead to less wastage of current and more efficient projectile propulsion but with increased circuit complexities. (ii) Second, by minimizing the wing length of the armature: Although the C-shape of the armature has many utilities, a larger wing length may cause more electromagnetic friction as it is directly proportional to the normal reaction that arises due to electromagnetic repulsion force of the rails as also discussed in Eq. (8). Minimization at a certain extent may lead to a decrease in this detrimental delay.

This paper demonstrates the pivotal role of the driving current pulse waveform that significantly impacts the acceleration efficiency of armature projectile. The reported experimental and simulation results unambiguously corroborate that a damped sinusoidal current feed results in much lower acceleration of an armature projectile as compared to an over-damped unidirectional current feed that flows in the barrel for adequate duration (i.e., until armature accelerates inside the barrel and then it finally comes out from end). The oscillatory nature of the sinusoidal current results in periodic acceleration of the armature projectile only when the oscillating current has a rising trend in either half cycle. Conversely, the unidirectional current feed results in consistent acceleration, leading to a higher final velocity of the armature projectile of similar mass. This fact is also substantiated by the estimation of the average force exerted on an 8 g armature projectile with a 220 kA peak current, i.e., ∼1.4 kN for the sinusoidal current feed and ∼3.83 kN for the unidirectional current feed, resulting in the estimated velocities of 489 and 931 m/s, respectively (corresponding acceleration efficiencies were ∼0.55% and 2%). To produce a more realistic estimate of the expected projectile velocity for a given mass and electrical energy input, the present simulation model may be further improvised by incorporating projectile mass loss effects on the kinematics of the projectile.

The authors thank all colleagues of Pulsed Power and Electromagnetics Division (PP and EMD), BARC Facility, Visakhapatnam, for their help and support during trials conducted on railgun with various armature projectiles.

The authors have no conflicts to disclose.

Dipjyoti Balo Majumder: Conceptualization (lead); Data curation (equal); Formal analysis (lead); Investigation (equal); Methodology (equal); Software (lead); Validation (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). Rishi Verma: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). J. M. V. V. S. Aravind: Data curation (equal); Investigation (equal). J. N. Rao: Data curation (equal); Investigation (equal). Manraj Meena: Data curation (equal). Lakshman Rao Rongali: Data curation (equal). Bijayalaxmi Sethi: Data curation (equal). Archana Sharma: Resources (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.