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European Journal of Scientific Research

ISSN 1450-216X Vol.41 No.2 (2010), pp.212-222

© EuroJournals Publishing, Inc. 2010http://www.eurojournals.com/ejsr.htm

Electro Magnetic - Thermal Analysis of Rail Gun Using Finite

Element Method

R. Murugan

Research Scholar, Anna University, Chennai 600 025

K. Udayakumar

Professor (Senior IEEE Member),Division of High Voltage Engineering

College of Engineering, Guindy, Anna University Chennai 600 025

Abstract

In this paper a coupled electromagnetic - thermal analysis of a rail gun system usingfinite element method is done to obtain the detailed information about the behavior of

specific rail gun component such as the rails and to predict overall performance of rail gun

system. Eddy current field solver is chosen to perform electromagnetic analysis andthermal field solver is chosen to perform thermal analysis. The electro magnetic analysis is

done to calculate inductance gradient of the rail, magnetic flux density between the rail,

current density distribution over the rail cross section and repulsive force acting on the rail.

Thermal analysis is done to calculate the temperature rise in a rail. The electro magneticloss found from the eddy current solver is coupled with thermal field solver and the

temperature rise in the rails is calculated. The inductance gradient of the rail, magnetic flux

density between the rail, current density distribution over a rail cross section repulsive forceacting on the rail and temperature rise in a rail mainly depend on rail dimensions. This

paper investigates how the rail dimensions affect these parameters using finite element

method. For various values of rail dimensions the above parameter values are calculatedand graphs are plotted. The performance of rail gun is discussed using the obtained graphs.

Keywords: Current density, Finite element method, Inductance gradient, Temperature

1. IntroductionThe rail gun is a device, which converts the electrical energy into mechanical energy for accelerating

the projectile to hypervelocity. It uses the magnetic field generated by the rail current to accelerate the projectile between the conducting rails. It consists of two parallel conductors, the rails that are bridged

by a non-ferromagnetic conductor called a projectile. Creating a current loop that flows from some

large power source down one rail, across the projectile, and back up the other rail fires the rail gun.This current loop induces enormous magnetic field, which, in turn, pushes the armature down the rails

with a force proportional to the magnitude of the current (I), the separation distance (L) of the rails, and

the magnetic field strength (B).



Electro Magnetic - Thermal Analysis of Rail Gun Using Finite Element Method 213

Figure 1: A simple graphic representation of rail gun

A simple graphic representation is as shown in Fig. 1, the current flows through one rail, passes

through the projectile perpendicular to the rails, and passes through other rail in parallel. As a result,

the two rails produces the magnetic field and an intercepting field produced by the armature. The railsrepel each other and they both in turn repel the armature. Since rails are both fixed the net result is a

propulsive force on the armature, which will be accelerated by electromagnetic means. The rails needto withstand enormous repulsive forces acting on it during the firing, and these forces intern will tend

to push them apart and away from the projectile. So the rails need to withstand this without bending,

and must be very securely mounted. The design of rail gun depends on the inductance gradient of therail, magnetic flux density between the rails, current density over rail cross section, repulsive force

acting on the rails and temperature distribution in the rail. The magnetic flux density between the rails

and inductance gradient of the rails play an important role in the performance of rail gun and itdetermines directly the force that accelerates the projectile. Any pairs of current carrying parallel

conductor experience to repulsive or attractive force depending upon the current directions. The same

force, which accelerates the projectile, also works to separate the rail pair. The separation force is notexactly calculated as short acceleration pulse current produces non-linear effect and uneven currentdistribution. As the rail gun is supplied with large currents (10

5to more than 10

6), which rise very

quickly and drives the armature down the rails. The thermal energy generated by the high current flow

through rail and armature change electrical and thermal property of the material. Since the time isshort, the current does not penetrate the rails or armature completely. The current density distribution is

not uniform over the cross section of each conductor. In this case, the current is distributed more near

the surface of each conductor. This makes the electromagnetic analysis of the rail gun extremelycomplex. Hence in order to gain a quantitative understanding of these parameters it is desirable to

calculate them well in advance. In general these values are affected by number of parameter such as

velocity of the moving armature, armature geometry, rail geometry, rail dimension, armature material

and rail material [1]. Previously, various mathematical models and codes were developed to computethe inductance gradient of the rail, magnetic flux density between the rails, current density over a rail

cross section, repulsive force acting on the rail and temperature rise in a rail [2-6]. Also researchers

focused on computation of these parameters [7-12] and measurement of it [14-15]. The objective of this investigation is to determine the values of these parameters for various rail dimensions using

coupled electromagnetic – thermal analysis. Finite element method is employed to compute these

parameter values. Graphs are plotted for the calculated values these parameters and the performance of rail gun is discussed using the obtained graphs. These results could be used in rail gun design.



214 R.Murugan and K.Udayakumar

2. Governing EquationsThe inductance gradient, magnetic flux density between the rail and repulsive force acting on the rail iscalculated using Eddy current solver. It calculates the magnetic field distribution in a space derived

from the Maxwell equations. In this analysis the magnetic vector potential (A) is first obtained from

the following relationships

∇X (1/μr (∇xA)=(σ + jω) (- jωA -∇ϕ) (3)

Where

A is the magnetic vector potential

ϕ is the electric scalar potential

μ is the magnetic permeability

ω is the angular frequency at which all quantities are oscillating

ε is the relative permittivityThe following section shows how this equation is developed from Maxwell equations

The eddy current solver solves the time harmonics electromagnetic field governed by the

following Maxwell equations

∇XH = J + ∂D/∂t (4)

∇XE=-∂B/∂t (5)

∇. D=ρ (6)∇. B=0 (7)

WhereE is the electric field

H is the magnetic field intensity

D is the electric field displacement

ρ is the charge densityJ is the current density

Using the following relation

H=μB (8)

D=εE (9)

J = σE (10)∂/∂t=jωfor time varying field (11)

The Maxwell equations are reduced to

∇ X (1/μr B) =(σ E + jεω E) (12)

∇ X E = - jωB (13)

∇.εE = ρ (14)

∇. B= 0 (15)

The eddy current solver actually solve for the magnetic vector potential that is given by

∇ x A = B (16)

Substituting the equation 16 in equation 12 the result is

∇ x (1/μr(∇ x A) = (σ E + jεω E) (17)

The solution for E in terms of A is given by

E = jωA - ∇φ (18)

Substituting equation 18 in 17 the result is

∇ x (1/μr (∇ x A) =(σ + jω) (- jωA -∇ϕ) (19)

This is the equation for eddy current solver, which is used to calculate the magnetic vector potential and scalar potential

After calculating the magnetic field distribution in a space, the inductance gradient of the rail is

calculated from the average energy stored in a rail.The average energy stored in the rail is given by

Eav = ∫ B.H*dv (20)




Since the instantaneous energy of system is

Einst = 1/2 Li2

(21)

Where i is the instantaneous value of currentInstantaneous value of current related to peak current is given by

i = I (cosωt+θ) (22)The average value of energy is then calculated by integrating the instantaneous energy

Eav= 1/2π ∫ Einst dωt = ∫ 1/ 2π ∫ 1/2 L (I (cos ωt+θ))2

dωt (23)

From this equation, the average energy stored in rail is equal toEav = L/4 (I peak)

2(24)

Then inductance gradient is given byL = 4 Eav/ (I peak)

2(25)

Which is used to calculate the inductance gradient of the rail.

The force acting on the rail can be calculated by using the relation

F=∫ JXBdv (26)Where J is the current density

B is the magnetic flux densityThis equation is used to calculate the repulsive force acting on the rail.

3. Thermal AnalysisUsing the obtained current distribution form the Eddy current analysis we can find the power loss

which is equivalent to thermal energyQ = J.J /σ (27)

Then the temperature distribution is obtained in the steady state equation

∇. (K ∇ T) = -Q (28)Where K is thermal conduction

This is the equation that the thermal field solver uses to calculate the temperature rise in a railfor a given rail geometry

4. Electromagnetic Analysis4.1 Problem Statement and Simulation Results

The inductance calculation of the two parallel and long conductors with uniformly distributed current

over the conductor cross section can be calculated easily. In a practical case, a very high magniture andshort duration current pulse is applied to the rail gun. Then the current is not uniform over the cross

section and is distributed in a very thin layer near the surface of each conductor. This makes theelectromagnetic analysis for a given rail gun geometry extremely complex. To simulate the rail gun in

this case transient time analysis must be used. Another way is the time harmonic or ac method in the

high frequency limit. In this method, all the current is distributed over the surface of the conductor.

This is a good approximation for rail guns with good conductor. In general case, the rail gun is a 3 Ddevice. But assuming that its barrel is infinitely long, the electromagnetic behavior can be analyzed

with the 2 D finite element models for the cross section of the barrel perpendicular to the longitudinaldirection. The cross sectional view of circular rail as shown in Fig.1.




Figure 2: Cross section of rail gun

This structure is commonly used in rail guns. The rails are of copper with conductivity equal to5.8 X 10

8S/m, and carrying a A.C current of 350 kA with higher frequency limit. Different ratios of

rail thickness to rail separations (E/S), for different rail separations (S) and opening angle (θ) of railgun are taken for modeling. Eddy current solver is chosen to perform electromagnetic analysis. It uses

finite element method to calculate the magnetic field distribution in a space for a given rail geometry.The current density distribution, magnetic flux density between the rails, inductance gradient of the

rails and repulsive force acting on the rails are computed for different values of E/S for different S and

θ. The rail separation is varied from 1cm to 4cm in step of 1cm and opening angle of rail varied from

25ο

to 45ο

increments of 10ο. The ratio of E/S is varied from 1 to 0.1. The Current density distribution

over a rail gun and the magnetic flux density distribution between the rails obtained from the

simulation for various values of E/S for various rail separation and opening angle Fig 3 and Fig 4.Fig.3 obviously shows that decreasing the ratio of E/S, rail separation and increasing the opening angle

of rail cross-section causes increase in the current density distribution. Moreover, the current is




Figure 3: Current density distribution over a rail cross section over a rail cross section

E/S =1, S= 4cm, θ = 45ο

E/S =0.9, S= 4cm, θ = 45ο

E/S =1, S= 3cm, θ = 45ο

E/S =1, S= 4cm, θ = 35ο




Figure 5: Inductance gradient value for various rail dimensions

(a) Rail separation S = 4cm (b) Rail separation S = 3cm

(c) Rail separation S = 2cm (d) Rail separation S = 1cm

(e) Rail opening angle θ = 45ο

(a) Rail separation S = 4cm (b) Rail separation S = 3cm

(c) Rail separation S = 2cm (d) Rail separation S = 1cm




Distributed in a very thin layer near the surface of rail edges so this phenomenon produces a hotspot that fuses the rail edges. Fig.4 shows that the magnetic flux density increases with decreasing the

rail separation and the ratio of E/S with increasing the opening angle of the rail. The calculated values

of inductance gradient and repulsive force acting on the rail against various rail dimensions are plottedand shown in Fig.5 and Fig.6. From the graph it is observed that the value of the inductance gradient

increases with an increase in θ and decrease in E/S. Also it is observed that the inductance gradientvalues increases as the rail separation decreases for the same ratio of E/S and opening angle of the rail.

In the case of repulse force acting on the rail, the value of force increases as the ratio of E/S and θ values are decreases. Also it is observed that the inductance gradient value increases weakly as the E/S

ratio is varied from 1 to 0.5 and hence it is not advisable to choose the ratio of E/S in these ranges

because in these ranges the inductance gradient value increases weakly thereby making the performance of rail gun poor. Hence the optimum ratio of E/S is less than 0.5.But in these ranges the

maximum value of current density and repulsive force acting on the rails are very high. Therefore

highly tensile materials with very high conductivity have to be chosen in these ranges.

5. Thermal Analysis5.1 Problem Statement and Simulation Result

Thermal field solver is chosen to perform thermal analysis over a given rail gun geometry. Eddy

current solver is used to assign the thermal source to the rail. It calculates electro magnetic loss produced in the rail due to the ohmic resistance of the rail. These losses are coupled to the thermal field

solver to calculate the temperature rise in a rail. The thermal conductivity of the copper rail is 400

W/mK. It is assumed that there is no heat transfer between the conductors and the surroundingmedium. The surface temperature of the rail is assumed to be room temperature. Melting point of

copper material is 1111degree centigrade. Simple graphical representation of electro - thermal coupled

rail gun model shown in Fig.6 Different ratios of rail thickness to rail separation (E/S) for various rail

separations (S) and opening angle (θ) of rail gun are taken for modeling.

Figure 6: Cross section of Electro thermal coupled model rail gun

Using the eddy current solver electromagnetic losses produced in the rail due to ohmic heat is

obtained for different ratios of E/S, opening angle the rail and rail separation. The electromagnetic

losses obtained from the simulation for the various ratios




E/S =1, S= 4cm, θ = 45ο E/S =1, S= 4cm, θ = 45ο

E/S =0.9, S= 4cm, θ = 45ο E/S =0.9, S= 4cm, θ = 45ο

E/S =1, S= 3cm, θ = 45ο E/S =1, S= 3cm, θ = 45ο

E/S =1, S= 4cm, θ = 35ο E/S =1, S= 4cm, θ = 35ο

of E/S for various rail separation and θ in Fig.7 From the figure it is observed that as the ratio of E/S,rail separation and opening of the rail cross sections decreases the electromagnetic loss in a rail is

increased. These losses are coupled with thermal field solver and the temperature rise in the rail is

calculated. The temperature rise in a rail crosses section obtained from the simulation for various raildimensions are shown in Fig.8. 7 From the figure it is observed that as the ratio of E/S, rail separation

and opening of the rail cross-sections decreases the electromagnetic loss in a rail is increased. Also it is

observed that the temperature value increases as the rail separation decreases for the same ratio of E/S

and opening angle of the rail cross sections.

6. ConclusionIn this paper, coupled electromagnetic – thermal field solver is used to investigate the effect of raildimensions on the maximum current density over a rail cross section, magnetic flux density between

the rail, repulsive force acting on the rail, inductance gradient of the rail and rise in temperature over a

rail cross section. Two dimensional finite element simulations are performed for various ratios of E/Sfor various values of rail separation and opening angle of the rail cross section. The maximum current

density, magnetic flux density, repulsive force and inductance gradient values are calculated using

eddy current solver and the graphs are plotted. Electro magnetic losses found from Eddy current solver is coupled with thermal field solver and temperature rise in a rail is calculated and the graph is plotted.

The performance of the rail gun was discussed using the obtained graphs. It is concluded that the value




of the inductance gradient increases with an increase in θ and decrease in E/S. Also it is observed thatthe inductance gradient values increases as the rail separation decreases for the same ratio of E/S and

opening angle of the rail. In the case of repulse force acting on the rail, the value of force increases as

the ratio of E/S and θ values are decreases. Also it is observed that the inductance gradient value

increases weakly as the E/S ratio is varied from 1 to 0.5 and hence it is not advisable to choose the ratioof E/S in these ranges because in these ranges the inductance gradient value increases weakly thereby

making the performance of rail gun poor. Hence the optimum ratio of E/S is less than 0.5.But in these

ranges the maximum value of current density and repulsive force acting on the rails and temparaturerise over rail cross section are very high. Therefore highly tensile materials with very high conductivity

have to be chosen in these ranges.

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inductance gradient” IEEE Transaction on Magnetic Vol.35 no.1, pp. 413 – 416 March 1999

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[11] L.Patch,J.M. Comstock,Y.C.Thfo and F.J.Young “ Rail gun Barrel design and Analysis” IEEETransaction on Magnetic Vol.Mag.20 No.2, pp. 360 – 363 March 1984

[12] S.P.Atekinson “The use of finite element analysis technique for solving rail gun problem” IEEE

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[13] Mohammad soleimani and Asgar Keshtkar “ A solution in Inhomoheneous 3D rail gun” IEEETransaction on Magnetic Vol.28 no.2, pp. 377 – 340 March 2000

[14] Alxander E.Zielnski and Calvin D. Le “ In bore elecatric and Magnetic field environment “

IEEE Transaction on Magnetic Vol.35 no.1, pp. 457 – 462 January 1999[15] Alxander E.Zielnski and Calvin D. Le “ Rail gun electric field: Theory and Experiment “IEEE

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