Yu-tao LOU*，Hai-yuan LI，Bao-ming LI

National Key Laboratory of Transient Physics，Nanjing University of Science and Technology，Nanjing 210094，Jiangsu，China

Research on proximity effect of electromagnetic railgun

Yu-tao LOU*，Hai-yuan LI，Bao-ming LI

National Key Laboratory of Transient Physics，Nanjing University of Science and Technology，Nanjing 210094，Jiangsu，China

The rails of electromagnetic railgun can be ablated by the temperature rise due to current concentration.The current distributions on the rails and armature are not only affected by the skin effect，but also inf l uenced by the proximity effect which is rarely mentioned.This paper illustrated the difference between skin effect and proximity effect，and the inf l uencing factors of proximity effect were investigated.Results show that the current is concentrated on the surface around rails due to the skin effect，and the proximity effect exacerbates the current density on the inner surfaces of rails.Decrease in distance from rails enhances the proximity effect，but has nothing to do with the skin effect，which also augments the rail resistance，resulting in temperature rise.It can explain the reason why the ablation is often detected in the small caliber railgun.Research results in this paper can provide support for design and optimization of electromagnetic railgun.

Electromagnetic railgun；Proximity effect；Skin effect；Ablation

1.Introduction

The electromagnetic railgun （EMG）is composed of rails，armature，insulatorsandcontainment［1-3］.Thearmaturecanbe spedupto2 km/sbyfeedingtheexcitingcurrenttothebreechof railgun ［4，5］.However，ablation occurs frequently on the rails during experiments.Researches show that one important reason of this phenomenon is temperature rise due to current crowding［6-8］.The current distributions on rails and armature are not only affected by the skin effect，but also inf l uenced by the proximityeffectwhichisrarelymentioned.Boththeeffectsplay important roles in the EMG performance and behavior.

In this paper，the difference between skin effect and proximity effect were discussed.The distribution of current density on rails was calculated in three cases.The f i rst case is for a single rail powered by a sinusoidal current pulse and only inf l uenced by the skin effect.The second case is for two adjacent rails，only left one is powered by sinusoidal current pulse. The third case is for two adjacent rails all powered by sinusoidal current pulse with opposite directions.In last two cases，the skin effect and proximity effect are both considered.

The proximity effect is not only affected by frequency like the skin effect，but also affected by the distance between rails. The current distribution due to proximity effect alters the internal impedance of rails，and inf l uences the EMG launch eff iciency f i nally.It is signif i cant and important to study the impact of proximity effect on electromagnetic railgun.

2.Formulations

The time-varying magnetic f i eld generated by the timevarying current induces the eddy current on the rails.It makes the current distribution toward the periphery of rails.This phenomenon can also be caused by a nearby rail，which is denoted as “proximity effect”［9，10］.

The modif i cations on the internal resistance and inductance due to proximity effect are not easily described by means of analytical formula.We have used a FEM code in the frequency domain to evaluate the EM f i eld of copper rails.Ansoft computer program was employed for calculation.The vector potential，current density，and magnetic f i eld were evaluated by Ansoft for each node of mesh.

The quantities are calculated by the following formulaswhere μ is the relative permeability；A is the vector magnetic potential；φ is the scalar potential；ω is the angular frequency；σ is the conductivity；ε is the dielectric constant；ITis the total exciting current；and J is the current density.

Fig.1.2D model and mesh of rail.

The A-φ method is employed to solve the wave Eqs.（1）and（2）.By obtaining the vector magnetic potential A and scalar potential φ of each node，the internal resistance R and inductance L can be solved by the following formulas

where Ipeakis the peak amplitude of harmonic current；V is the volume of conductor；and H is the magnetic f i eld intensity.In the program of Ansoft，the default length of a 2D model is 1m. Thus，the values of resistance gradient R‘and inductance gradient L'are equal to R and L，respectively.

Fig.2.Current distribution and magnetic f i eld at 100 kA and 200 Hz.

3.Simulations and results

3.1.Simulation parameters

To study the difference between skin effect and proximity effect，a simplif i ed two-dimensional model was established for simulation.The geometrical parameters of the model are that the width and height of rail cross-section are 10 mm and 20 mm，respectively.The rails are made of copper，and its conductivity σ is 5.8e7S/m and magnetic permeability is μ0.

A triangular mesh was used for the FEM simulation.The maximum length of mesh elements is 2 mm.The maximum surface triangle length of model is 1 mm with the constraint of the skin penetration depth δ expressed as

where f is the frequency.The mesh of model is depicted in Fig.1.

3.2.Current distribution and magnetic line

The exciting current is a sinusoidal current，of which amplitude is 100 kA and frequency is 200 Hz.To research the difference between skin effect and proximity effect，the current distributions of rails in three cases mentioned above were discussed.The simulated results are shown in Fig.2.

As shown in Fig.2 （a），for the f i rst case，the current tends to the periphery of rail，and the maximum and minimum current densities are 641 GA/m2and 338 GA/m2，respectively.The current density outside the rail is higher than that inside.

In the second case，another rail without exciting current is added.The distance of two rails is very short（d=0.01 mm）.As shown in Fig.2 （b），the current density concentrates on the adjacent side of two rails.The current density on the right rail only inf l uenced by the proximity effect has the same order as that on the left one.Due to the proximity effect，the magnetic lines on the left rail are compressed by the magnetic f i eld on the right rail.It leads to Jmax（690 GA/m2）on the left rail to be higher than that，which is 640 GA/m2，in the f i rst case.The proximity effect makes the right rail like a shield and increases the current density of nearby rail.

In the third case，both the rails were powered by sinusoidal current pulse with opposite directions for simulating the actual situation.The results are shown in Fig.2 （c）；the skin effect and proximity effect make the current more concentrated to the adjacent side.Both maximum current densities of two rails are higher than that in the second case.The current distributions and magnetic lines of two rails are symmetrical.

Fig.3 shows the current density on the lines which are the boundary of adjacent sides of two rails.Obviously，the current densities on lines A′-B′ and C′-D′are higher than those in the fi rst two cases.The maximum current density Jmaxis 713 GA/m2.

Through these calculations，we can learn that the skin effect makes the current concentrate to the periphery of rails，and the proximity effect exacerbates the current crowding on the adjacent side.

Fig.3.Current densities on boundary lines.

Fig.4.Resistance and inductance gradients versus distance.

3.3.Three factors of proximity effect

The exciting current and frequency still remain 100 kA and 200 Hz，respectively.When the distance between rails changes from 0.01 mm to 1000 mm，Jmaxis in the range from 713.4 GA/m2to 641.5 GA/m2.The current distribution on the rails is basically the same as that shown in Fig.2 for d=1000 mm.This is because that the proximity effect is weakened with the increase in distance，while the skin effect is not affected by the distance.It is well known that the internal impedances of rails are depended on the current distribution. Thus，the proximity effect inf l uences the impedance of rails at different distances.

Fig.5.Resistance and inductance gradients versus frequency.

Fig.6.Current density distributions at different current amplitude.

As shown in Fig.4，with the increase in distance between rails，the resistance gradient R'decreases from 252.2 μΩ/m to 202.8 μΩ/m，and the inductance gradient L'increases from 0.263 μH/m to 1.99 μH/m.R'reduces rapidly when d is in the range from 0.01 mm to 100 mm，and it basically remains unchanged when d is more than 100 mm.Therefore，the increase in the caliber of EMG not only improves the inductance gradient，but also reduces the resistance gradient greatly. It's signif i cant for launch eff i ciency of EMG.

The proximity effect is not only affected by the distance，but also affected by the frequency.The resistance and inductance gradients versus frequency for d=20 mm and Ipeak=100 kA is shown in Fig.5.R'increases from 185.4 μΩ/m to 426.4 μΩ/m，and L'decreases from 0.605 μH/m to 0.549 μH/m slowly. Compared with the inductance gradient，the resistance gradient is more sensitive to frequency.Thus，for EMG，a low frequency current can be used to not only improve the inductance gradient，but also reduce the resistance gradient greatly.

When distance and frequency are constant，the current density distribution does not change with the exciting current，as shown in Fig.6.Thus，the current amplitude has nothing to do with the proximity effect.

4.Conclusions

The research on current distribution of rails is very important for an electromagnetic railgun （EMG）.In this paper，the inf l uences of skin effect and proximity effect on current density distribution were analyzed.Through the comparison of three cases，it is verif i ed that the skin effect makes the current tend to the periphery of rails，and the proximity effect exacerbates the current crowding on the adjacent side.

To research how the proximity effect affects EMG，three impact factors，such as the distance between rails，and the frequency and amplitude of exiting current，were discussed. The results show that the proximity effect is aggravated by decreasing the distance between rails and increasing the frequency，leading to an increased resistance gradient and a decreased inductance gradient.This results in temperature rising，which is adverse to the EMG launch eff i ciency.The ablation of rails appears more frequently in small caliber railgun with high frequency exciting current.The research results in this paper can provide support for the design and optimization of electromagnetic railgun.

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Received 20 October 2015；revised 12 January 2016；accepted 25 January 2016 Available online 3 March 2016

Peer review under responsibility of China Ordnance Society.

*Corresponding author.Tel.13390905762.

E-mail address:louyutao1988@163.com （Y.T.LOU）.

http://dx.doi.org/10.1016/j.dt.2016.01.010

2214-9147/? 2016 China Ordnance Society.Production and hosting by Elsevier B.V.All rights reserved.

? 2016 China Ordnance Society.Production and hosting by Elsevier B.V.All rights reserved.