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Simulation of Electromagnetic Pulse Propagation Through Dielectric Slabs with Finite Conductivity Using Characteristic-Based Method

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Simulation of Electromagnetic Pulse Propagation Through Dielectric Slabs with Finite Conductivity Using Characteristic-Based Method PIERS ONLINE, VOL. 2, NO. 6, 2006 609 Simulation of Electromagnetic Pulse Propagation through Dielectric Slabs with Finite Conductivity Using Characteristic-based Method Mingtsu Ho Department of Electronic Engineering, WuFeng Institute of Technology Chia Yi 621, Taiwan Abstract| This paper presents one-dimensional simulation results of electromagnetic pulse propagation through various dielectric slabs. The material used in the numerical model is assumed to be homogeneous and isotropic such that its dielectric constant, relative permeability, and conductivity are not functions of the time and independent of the electric and magnetic ¯eld intensities. The computational results are generated by the characteristic-based method and compared in a side-by-side fashion. The e®ects of medium conductivity on the reflected and transmitted pulses are recorded and shown as time elapses. Also revealed is how the medium conductivity a®ects the electromagnetic pulses for dielectric slabs that are impedance matched to free space, i. e., "r = ¹r. The frequency-domain results are illustrated that are obtained through the application of FFT to the time-domain data. DOI: 10.2529/PIERS060825013017 1. INTRODUCTION The method of moment (MoM) and the ¯nite-di®erence time-domain (FDTD) technique have been the two most popular computational electrodynamics modeling techniques ever since they were proposed in 60 s [1, 2]. The latter can be formulated for either time- or frequency-domain analysis while the former is only in frequency-domain. Numerical simulation results always provide researchers a much more perceptive view of a variety of electromagnetic phenomena and therefore give researchers a better understanding of the problem of interest. This is in contrast to most textbooks where electromagnetic problems are highlighted in frequency-dependent formulations. Furthermore, through proper Fourier transformation one can easily maneuver the computational results, from time-domain transient data to frequency-domain spectrum analysis or vice versa. The characteristic-based method was proposed in mid-90 s and found to produce results in good agreement with data produced by FDTD [3]. Unlike MoM and FDTD, the present numer- ical method is an implicit numerical procedure, places all ¯eld components in grid center, solves Maxwell's equations by balancing all fluxes across cell faces within each computational cell, and yet pays the cost of recasting the governing equations from the Cartesian coordinates into the body- conforming coordinates. It is shown that the characteristic-based method can predict the reflection of electromagnetic ¯elds from moving/vibrating perfect conductor in one dimension [4] and the reflection/transmission of electromagnetic ¯eld propagation onto moving dielectric half space [5]. In order to investigate the e®ects of medium conductivity on the electromagnetic ¯eld as it propagates through the medium, one has to include the conductivity in the governing equations, Maxwell's equations. Luebbers et al. simulated the interaction of a Gaussian pulse plane wave with a dielectric slab of ¯nite thickness, constant relative permittivity and conductivity in one-dimension using FDTD [6]. In formulation, FDTD explicitly speci¯es the medium properties at grid node in accordance with the designation of the electric ¯eld and the magnetic ¯eld components that are interleaved in time and space. These medium properties are permittivity ("), permeability (¹), and conductivity (σ). The inclusion of the conductivity term in the governing equation is simply meaning a source term, ¡σE. In the FDTD expression, conductivity is multiplied by the numerical time step (¢t) and then combined with permittivity at each grid point. In the characteristic-based algorithm, it is an extra quantity added to the calculated flux di®erence for every cell inside the medium. Since the main objective of this paper is to simulate and demonstrate the computational re- sults of electromagnetic pulse propagation through di®erent conducting dielectric slabs using the characteristic-based method, the comparison between numerical methods will not be covered here. PIERS ONLINE, VOL. 2, NO. 6, 2006 610 The computational results are illustrated as follows. The e®ects of medium conductivity on the reflected and transmitted electromagnetic ¯elds are plotted as functions of the time. Especially, for the case when the medium is impedance matched to free space (the reflection coe±cient is zero in magnitude), we also demonstrate how conductivity a®ects the electromagnetic pulse by assum- ing this impedance matched medium is also conductive. Two groups of plots of the time-domain reflected and transmitted electric ¯elds and their spectra are shown and compared. 2. GOVERNING EQUATIONS Maxwell's equations are the governing equations of electromagnetic problems. When electromag- netic waves propagate onto a medium with ¯nite conductivity, Maxwell's equations become @B~ + r £ E~ = 0; (1) @t @D~ ¡ r £ H~ = ¡σE;~ (2) @t r ¢ D~ = 0; (3) r ¢ B~ = 0; (4) where E~ and H~ are the electric and magnetic ¯eld intensities, D~ and B~ are the electric and magnetic flux densities, respectively. The two constitutive relations are D~ = "oE~ and B~ = ¹oH~ with "o and ¹o are the permittivity and permeability of vacuum. Inside medium, these two expressions are given by D~ = "E~ = "r"oE~ and B~ = ¹H~ = ¹r¹oH~ with "r and ¹r are the dielectric constant and relative permeability of medium. Finally, symbol σ is the medium conductivity. If the medium is conductive, then σE~ represents the current density. Once more, in the numerical model we assume ", ¹, and σ are not functions of the time. Since the present work focuses on one-dimensional electromagnetic ¯eld propagation problem, we can only consider a two-dimensional formulation and make the following assumptions. In the numerical model, the electric ¯eld intensity is polarized to z-direction and has a peak value of unity, the incident pulse initially propagates in the positive x-direction and normally illuminates upon a dielectric slab that may be featured with a constant conductivity. As stated earlier, the Maxwell's equations have to be recast from the Cartesian coordinate system (t; x; y) into curvilinear coordinate system (¿; »; ´) and are written as @Q @F @G + + = ¡σE (5) @¿ @» @´ z T T where Q = Jq, F = J (»xf + »yg), G¯ = J (´¯xf + ´yg), q = [Bx;By;Dz] , f = [0; ¡Ez; ¡Hy] , ¯@(x; y)¯ g = [E ; 0;H ]T , and J is equal to ¯ ¯. The numerical formulation is derived by applying z x ¯ @(»; ´) ¯ the central di®erence to (5) and are written as Qn+1 ¡ Qn ± F ± G + i + j = ¡σE (6) ¢¿ ¢» ¢´ z where ¤ ¤ ¤ ±k( ) = ( )k+1=2 ¡ ( )k¡1=2 (7) is known as the central-di®erence operator. Symbols Qn+1 and Qn are the variable vectors of two successive time levels, ¢» and ¢´ represent the cell size between discrete points in the computa- tional domain, ¢¿ is the numerical time step, the subscript (k) of the central-di®erence operator represents the direction in the curvilinear coordinate system and the one-half integer index in (7) indicates that the flux vectors F and G are evaluated at the cell face. The flux vector splitting technique is then applied to (6) and the lower-upper approximate factorization scheme is employed for the solution of the system of linear equations. The boundary condition treatments in the present simulation simply assure that the tangential components of both electric and magnetic ¯eld intensities are continuous across the interface be- tween two media, and that the out-going ¯elds will not be bounced back into the computational domain from the outer computational layer. PIERS ONLINE, VOL. 2, NO. 6, 2006 611 3. CASES STUDIED The problem of interest is de¯ned in Figure 1 where the electric ¯eld intensity is shown. As illustrated the incident pulse is Gaussian in form and propagates toward the dielectric slab. The Gaussian pulse has a width of about 0.695 ns, a truncated level of 100 dB, a span of 2 meters, and a highest frequency content of about one GHz. The dielectric slab is 30 cm thick and may be speci¯ed by a ¯nite conductivity whose magnitude is normalized by ´0 for the purpose of clear demonstration. For clear demonstration we further assume that the reference time zero is when the peak of the incident pulse is at x = 1 m and set a constant time interval ¢t = 0:2 m/C where the symbol C is the light speed and exactly equal to 3 £ 108 m/s in the present simulation. This means the numerical electromagnetic pulse propagates precisely 0.2 meters during the time interval ¢t. Figure 1: De¯nition of the problem. Figure 2: Propagation of electromagnetic pulse through two dielectric slabs. Solid lines: σ = 1; Dotted lines: σ = 0. ; Figure 3: Propagation of electromagnetic pulse Figure 4: Propagation of electromagnetic pulse through two impedance-matched dielectric slabs. through two conducting dielectric slabs. Solid lines: Solid lines: σ = 1; Dotted lines: σ = 0. "r = 2, ¹r = 2; Dotted lines: "r = 4, ¹r = 1. The medium used in the simulation is assumed to be homogeneous and isotropic. In order to observe the e®ects of medium conductivity on the electromagnetic ¯elds, we further specify the properties of dielectric slabs as follows. They may be made of either non-magnetic material with "r = 4 and ¹r = 1 or material with "r = 2 and ¹r = 2 whose impedance is matched to the free space. 4. RESULTS For the purpose of easy and clear comparison, two sequences of plots of electric ¯elds for elec- tromagnetic pulse propagation through two di®erent dielectric slabs are given in Figure 2. It is PIERS ONLINE, VOL. 2, NO. 6, 2006 612 shown that the reflected electric ¯eld from conducting slab is stronger in magnitude than that from lossless slab, and that the attenuation of the transmitted ¯eld as it propagates through the lossy medium. Also noticed are that the reflected electric ¯eld bears a dragging tail due to the medium conductivity, and that the electromagnetic pulse is slow down by one half as it propagates through the dielectric slab.
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