1398 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 41, NO. 5, MAY 2013

Analyzing the Near and Far Field Using FiniteDifference and Finite Element Method

Mohammad Sajjad Bayati, Asghar Keshtkar, and Leila Gharib

Abstract— In this paper, the railgun is assumed to be anantenna and simulated with the finite element method (FEM)and the finite difference time domain (FDTD) method. For FEManalysis, the caliber of the railgun is 10 × 10 mm, rail thicknessis 10 mm, and rail length is 400 mm. At first, the H-field onthe cross section of the rails is computed. Next, power radiationand the maximum value of the E-field in the near and far fieldare obtained as a function of frequency. Polar radiation patternsversus frequency (1 kHz–1 MHz) in the near and far-field regionsare calculated as a function of the angle. For FDTD, we assumethat the railgun operates in the far-field region, and radiationpatterns are computed in this region. The caliber of the railgunis 15 × 19 mm, rail thickness is 10 mm, and length is 1000 mm.Directivity has been computed versus frequency (100 Hz–1 MHz)in the far-field region. The H-field is calculated versus frequencyand the energy is computed as a function of time. For bothmethods, the railgun included copper rails and an aluminumprojectile.

Index Terms— Far field, finite difference time domain, finiteelement method, near field, railgun.

I. INTRODUCTION

ASIMPLE railgun in an electromagnetic launcher systemconsists of two parallel rails and a solid projectile. The

current loop in the railgun is closed by the solid projectile.By solving Maxwell’s equations, with boundary conditionsapplied, important quantities can be calculated. In recent years,extensive research has been done on computing quantities forrailguns using numerical methods, e.g., inductance gradient[1]–[3], current density [3]–[5], force [6], thermal distribution[7], [8], and pressure on the rails [9], which are essentialparameters of a railgun. Of course, the inductance gradientplays an important role in the performance of a railgun, andit can be computed by using numerical methods such as thefinite element method (FEM) [1], [3], [4], [9], finite differencetime domain (FDTD) [5], and boundary element method [10].The calculation and measurement of the H -field can be donewith FEM using the H -dot sensors around the rails [11]. Thefield in the neighborhood of the railgun is computed for thecase of a constant current [12].

Manuscript received February 6, 2013; revised February 20, 2013; acceptedFebruary 26, 2013. Date of publication April 9, 2013; date of current versionMay 6, 2013. This work was supported by grant 1/56020 from the ImamKhomeini International University.

M. S. Bayati is with the Department of Electrical Engineering, RaziUniversity, Kermanshah 08314274535, Iran (e-mail: s.bayati@gmail.com).

A. Keshtkar and L. Gharib are with Imam Khomeini InternationalUniversity, Ghazvin 31546-15157, Iran (e-mail: akeshtkar@gmail.com;gharib_leila@yahoo.com).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPS.2013.2251480

Fig. 1. Simple railgun. FE mesh configuration of a railgun.

Fig. 2. Mesh generation of the railgun for finite difference solution.

This paper attempts to analyze a railgun that is able toradiate low and high frequencies. In this paper, we analyzethe railgun radiations that vary sinusoidally in time.

We model the railgun as a current loop and simulateit with 3-D FEM and FDTD. Assuming that the railgunis operated in the far-field region, the radiation patternsare computed in the far-field region where the transmittedwave of the transmitting current loop resembles a sphericalwave from a point source that only locally resembles auniform plane wave. Figs. 1 and 2 show the mesh gen-eration of the railgun for FEM and FDTD, respectively.For 100 Hz−1 MHz, the wavelength is 300 m−3 km.In the region near the railgun, R < λ or R < 300 m, andwe choose R as 10 cm. For the region far from the railgun,R is much larger than λ. We implement FEM with the HFSSsoftware and FDTD with the CST software.

0093-3813/$31.00 © 2013 IEEE

BAYATI et al.: ANALYZING THE NEAR AND FAR FIELD USING FDTD AND FEM 1399

Fig. 3. Magnetic field on rail cross section.

II. FINITE ELEMENT METHOD IMPLEMENTATION

A. Governing Equations

All electromagnetic phenomena can be mathematicallydescribed by a set of Maxwell’s equations. After the definitionof the grid cell or mesh generation, we start with the followingequation:

�E(x, y, z) =∫

s(≺ jwμ0 Htan G � + ≺ Etan × ∇ G �+ ≺ En × ∇ G �) ds (1)

where E is the electrical field intensity, Htan is the tangentmagnetic field intensity, Etan is the tangent electrical fieldintensity, En is the normal electrical field intensity, G is theGreen function in free space, f is the frequency, and μ0 isthe magnetic permeability in free space. Green’s function ofthe current element for free space is

G = exp(− jk |R|)|R| (2)

where R is the distance from any point on the source to theobservation point, and k is a constant of propagation in freespace. The magnetic field intensity can be calculated from

�H = 1

η�E × R (3)

where η is the characteristic impedance for free space. Theaverage power radiation can be calculated from the followingequations:

Pav = 1

2Re

(∫s

�E × �H ∗.n ds

). (4)

For free space η is 120π .

B. FEM Results

The caliber of the railgun is 10 × 10 mm, rail thickness is10 mm, and rail length is 400 mm. Fig. 3 shows the magneticfield on the rail cross section. The electrical field computed in rdirection for far field and near field is shown in Figs. 4 and 5.Polar radiation patterns versus frequency (1 kHz, 10 kHz,100 kHz, and 1 MHz) in the near- and far-field region are cal-culated as a function of the angle and are shown Figs. 6 and 7.Figs. 8–10 show the radiation power and maximum values ofthe E-field in near and far field as a function of frequency.

Fig. 4. Space view of Er in the far field at f = 5 kHz.

Fig. 5. Space view of Er in the near field at f = 5 kHz.

III. FINITE DIFFERENCE TIME DOMAIN IMPLEMENTATION

The simple electromagnetic launcher includes two parallelrails and a solid projectile. The cross section of the rails ish × w, the length of the rail is d , and the cross section of theprojectile is h × s. The railgun dimensions are h = 15 mm,w = 10 mm, s = 19 mm, and d = 1000 mm [11]. For thiscase study, the maximum value of the H -field as a functionof frequency, radiation pattern, and directivity in r and ϕ aredirectly computed using the equations below. The results areshown in the accompanying figures.

A. Governing Equations

All electromagnetic phenomena can be mathematicallydescribed by a set of Maxwell’s equations. After the definitionof a grid cell or mesh generation, we start with Faraday’s lawin the integral form∮

∂ A

�E(�r , t). d�s = −∫∫

A

∂

∂ t�B(�r, t). d �A (5)

where E is the electrical field intensity and B is the mag-netic flux density. Maxwell’s second equation describes thenonexistence of magnetic charges in the integral form∫∫

∂V

�B(�r , t). d �A = 0. (6)

1400 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 41, NO. 5, MAY 2013

(a) (b)

(c) (d)

Fig. 6. Radiation pattern for θ = 0° and 90° in the far field. (a) f = 1 kHz.(b) f = 10 kHz. (c) f = 100 kHz. (d) f = 1 MHz.

(a) (b)

(c) (d)

Fig. 7. Radiation pattern for θ = 0° and 90° in the near field (a) f = 10 kHz.(b) f = 100 kHz. (c) f = 1 MHz. (d) f = 10 MHz.

Ampere’s law in the integral form is given by

∮∂ A

�H (�r , t). d�s =∫∫

A

(∂

∂ t�D(�r , t) + �J (�r , t)

). d �A (7)

Fig. 8. Radiation power versus frquency.

Fig. 9. Maximum values of E-field in near field as function of frequency.

Fig. 10. Maximum values of E-field in far field as function of frequency.

where H is the magnetic field intensity, D is the electricalflux density, and J is the current density∮

s

�D.n ds =∫∫∫

vρe dv. (8)

For numerical calculations in the time domain, it is alsonecessary to discretize the time axis of the electromagneticlauncher process

f (t), t0 ≺ t ≺ tnf (tk), t0 ≺ tk ≺ tn, k = 0, . . . , n. (9)

After grid cell or mesh generation and using Maxwell’sequations, three case studies can be simulated.

B. FDTD Results

Fig. 11 shows the magnetic field intensity around the railsand in comparison with the results in [11]. Fig. 12 shows the

BAYATI et al.: ANALYZING THE NEAR AND FAR FIELD USING FDTD AND FEM 1401

Fig. 11. Magnetic field intensity as a function of frequency. Result in thispaper (•). Results of [11] (�) and (∗).

(a) (b)

(c) (d)

Fig. 12. Directivity of railgun versus frequency. (a) f = 100 Hz.(b) f = 1 kHz. (c) f = 100 kHz. (d) f = 1 MHz.

directivity for 100 Hz, 1 kHz, 100 kHz, and 1 MHz. Fig. 13shows the directivity in the ϕ direction for 100 Hz and 1 MHz.Fig. 14 shows the field energy as a function of time.

IV. RESULTS AND DISCUSSION

The propagation patterns of near E-fields depicted in thepolar system (Figs. 6 and 7) for angles of 0° and 90°, representthe increase of the field magnitude with respect to the increaseof frequency. Besides the magnitude variation, the facts thatcan be investigated from these diagrams are as follows. First,the resulting pattern is circular for the angle 0°. In other words,the field magnitude for θ = 0° is invariable with respect to thechanges of ϕ. Second, the pattern transforms to a petal form forθ = 90°, and increases the frequency. Third, the reason whythe near E-field pattern is not depicted for 60° < θ < 120°

(in spite of θ = 90° where the point of origin refers to) is thatthe launcher is located in this area. Since all components of

(a)

(b)

Fig. 13. Directivity in ϕ direction. (a) f = 100 Hz. (b) f = 1 MHz.

Fig. 14. Field energy as a function of time.

the launcher are conductive, the electrical field is zero insidethe ranges in which the launcher is located. Consequently, themagnitude of near E-filed is also zero in this range. However,this is not the case for the propagation patterns in the far field,depicted in Fig. 7, for θ = 0° and θ = 90°. Besides that, thefar E-field magnitude increases as the frequency increases.There are two other facts investigated from these figures: first,the pattern for the angle of zero is circular, which means thatthe magnitude of far E-field for θ = 0° is independent of thevariation of ϕ. Second, the pattern for θ = 90° is bilaterallysymmetric with respect to the direction of ϕ = 45°. As shownin Fig. 8, the propagated power starts with an abrupt increase atlow frequencies, but its rate of variation drastically decreases

1402 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 41, NO. 5, MAY 2013

at frequencies above 10 MHz. However, the magnitude of thepropagated power is small at low frequencies. As depictedin Fig. 9, the maximum magnitude of the launcher nearE-field increases at low frequencies abruptly. However, its rateof variation decreases at frequencies above 10 MHz as well.Nevertheless, the maximum magnitude of the near E-field inthis case is much higher than the cases shown in Fig. 9 andthe maximum magnitude of the far E-field shown in Fig. 10,both in low and high frequencies.

V. CONCLUSION

In this paper, we studied railgun radiation based on theantenna theory. According to FEM and FDTD results, the mag-nitude of far E-field and the radiation fields at low frequenciesare incomparable to the environmental noise. Considering theresults of FEM, the equipment located inside the near-fieldradius can be affected because of the high amplitude of thefield.

REFERENCES

[1] B.-K. Kim and K.-T. Hsieh, “Effect of rail/armature geometry on cur-rent density distribution and inductance gradient,” IEEE Trans. Magn.,vol. 35, no. 1, pp. 413–416, Jan. 1999.

[2] A. Keshtkar, S. Bayati, and A. Keshtkar, “Derivation of a formula forinductance gradient using IEM,” IEEE Trans. Magn., vol. 45, no. 1,pp. 305–308, Jan. 2009.

[3] M. S. Bayati, A. Keshtkar, and A. Keshtkar, “Transition study of currentdistribution and maximum current density in railgun by 3D FEM andIEM,” IEEE Trans. Plasma Sci., vol. 39, no. 1, pp. 13–17, Jan. 2011.

[4] J. F. Kerrisk, “Current diffusion in rail-gun conductors,” Los AlamosNational Lab., Alamos, NM, USA, Tech. Rep. LA-9401-Ms, Jun. 1982.

[5] J. Gallant, “Parametric study of an augmented railgun,” IEEE Trans.Magn., vol. 39, no. 1, pp. 451–456, Jan. 2003.

[6] M. S. Bayati, A. Keshtkar, and A. Keshtkar, “Thermal computation inrailgun by hybrid time domain techniqe 3D FEM and IEM,” IEEE Trans.Plasma Sci., vol. 39, no. 1, pp. 18–21, Jan. 2011.

[7] K. T. Hsieh, “Hybrid FE/BE implementation on electronimechanicalsystems with moving conductors,” IEEE Trans. Magn., vol. 43, no. 3,pp. 1131–1133, Mar. 2007.

[8] K. T. Hsieh, S. Satapathy, and M. T. Hsieh, “Effects of pressuredependent contact resistivity on contant interfacial condition,” IEEETrans. Magn., vol. 45, no. 1, pp. 313–318, Jan. 2009.

[9] K. T. Hsieh, “A Lagrangian formulation for mechanically, thermallycoupled electromagnetic diffusive processes with moving conductors,”IEEE Trans. Magn., vol. 31, no. 1, pp. 604–609, Jan. 1995.

[10] A. Keshtkar and S. Bayati, and A. Keshtkar, “Effect of rail’s materialon railgun inductance gradient and losses,” in Proc. 14th EML Symp.,Jun. 2008, pp. 130–133.

[11] W. O. Coburn, C. Le, D. J. DeTroye, and G. E. Blair, “Electromagneticfield measurment near a rail,” IEEE Trans. Magn., vol. 31, no. 1,pp. 698–703, Jan. 1995.

[12] W. G. Soper, “Electromagnetic fields near rail guns,” IEEE Trans.Antenna Propag., vol. 37, no. 1, pp. 128–131, Jan. 1989.

Mohammad Sajjad Bayati was born in Sonqor,Iran, in 1979. He received the B.Sc. degree inelectrical engineering from the University of Tabriz,Tabriz, Iran, in 2002, the M.Sc. degree in telecom-munication field and wave engineering from theSahand University of Technology, Sahand, Iran, andthe Ph.D. degree in electrical engineering from theUniversity of Tabriz in 2011.

He is an Assistant Professor with the Departmentof Electrical Engineering, University of Razi, Ker-manshah, Iran. His current research interests include

electromagneticism, mathematics in electromagneticism, and electromagneticlaunchers and antennas.

Asghar Keshtkar was born in Ardabil, Iran, in1962. He received the B.Sc. degree in electricalengineering from Tehran University, Tehran, Iran,the M.Sc. degree in electrical engineering from theUniversity of Khaje-Nasir, Tehran, and the Ph.D.degree in electrical engineering from the Iran Uni-versity of Science and Technology, Tehran, in 1989,1992, and 1999, respectively.

He is currently a Professor with the Facultyof Engineering and Technology, Imam KhomeiniInternational University, Ghazvin, Iran. His current

research interests include electromagneticism, bioelectromagneticism, andelectromagnetic launchers and antennas.

Leila Gharib photograph and biography are not available at the time ofpublication.