Scattering of Electromagnetic Waves from Vibrating Perfect Surfaces: Simulation Using Relativistic Boundary Conditions

Transcription

1 74 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August Scattering of Electromagnetic Waves from Vibrating Perfect Surfaces: Simulation Using Relativistic Boundary Conditions Mingtsu Ho WuFeng Institute of Technology, Taiwan Abstract One-dimensional numerical simulation of electromagnetic waves scattered from vibrating perfect conductor is reported in this paper. The computational results are generated by the method of characteristics with the application of the relativistic and characteristic variable boundary conditions. The perfect conductor may oscillate either in zigzag or sinusoidally. For easy observation of oscillatory effects on the reflected electromagnetic waves, objects are set to vibrate at a very high frequency with constant amplitude such that the extreme instantaneous velocity is about one tenth of the speed of light. The numerical calculations show that the reflected electric fields bear the oscillatory characteristics of the vibrating perfect conductor and that the change in magnitude are in agreement with the theoretical Doppler shift values. Introduction For the past half-century the two most widely used computational methods for electromagnetic problems have been the method of moment (MoM) and the finite-difference time-domain (FDTD) technique. They provide numerical results giving researchers better understanding oarious problems. The method of characteristics (MOC) was proposed and initially applied to the fluid dynamic problems in the early 80s by Whitfield and Janus [1]. Shang applied explicit central difference technique in a characteristic-based algorithm to approximate the time-domain Maxwell curl equations in the early 90s [2]. Soon it was formulated in an implicit form and found to produce results in good agreement with data produced by FDTD [3]. This numerical approach has advantages such as to be able to solve problems involved with objects having curvature without staircasing the object surface and to perform well when the cell size is changing with time without interpolation techniques [4]. Unlike MoM and FDTD, MOC places all field variables in the center of the grid cell and solves Maxwell s equations by balancing all fluxes across cell faces within each computational cell, which enables the method to easily handle time-varying cells. In many circumstances, dealing with electromagnetic scattering problems either from moving and/or vibrating objects becomes necessary and important wherever such effects cannot be ignored. The one-dimensional analytical theories of scattering of EM fields by a moving boundary [5] and that by linearly vibrating objects [6] have been studied. The following conclusions were made. The oscillatory behavior of target impress upon the scattered fields. The scattered EM waves from oscillating target are altered in phase and magnitude when compared with that of a stationary target, which obey the double-doppler shifts. Though inspired by many studies, the present work focuses only on the application of the MOC numerical approach to one-dimensional simulation of electromagnetic waves scattered from perfect electrical conductors (PEC) oscillating either in zigzag or sinusoidally. The scheme accuracy is investigated by comparing the computational results with the theoretical Doppler shift values. Governing Equations and Boundary Conditions The governing equations for electromagnetic problems in source-free region are the Maxwell s equations. Since the present algorithm numerically solves Maxwell s equations in one-dimension, we can only consider a two-dimensional formulation and make the following arrangements. In the model, the electric field intensity is z- polarized and normalized to unity, the incident waves initially propagate in the positive x-direction and normally illuminate upon a perfect conductor that is vibrating at a very high frequency. The Maxwell s equations are transformed from the Cartesian coordinate system (t, x, y) into a time-invariant curvilinear coordinate system (τ, ξ, η) and rewritten as: Q τ + F ξ + G η = 0 (1)

2 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August where Q = Jq, F = J(ξ x f + ξ y g), G = J(η x f + η y g), q = [B x, B y, D z ] T, f = [0, E z, H y ] T, g = [E z, 0, H x ] T, and symbol J is the Jacobian of the inverse transformation and equal to x y ξ η x y. The implicit formulation is derived first by applying the central difference to (1) and written η ξ as where Q n+1 Q n τ + δ if ξ + δ jg η = 0 (2) δ k ( ) = ( ) k+1/2 ( ) k 1/2 (3) is known as the central-difference operator. The superscripts n and n+1 on variable vector Q are the two successive time levels, the subscripts (i) and (j) in (2) and (k) in (3) represent various directions in the curvilinear coordinate system, and the one-half integer index in (3) indicates that flux is evaluated at the cell face along one particular direction. The flux vector splitting technique and the Newton iterative method are then applied to (2) followed by the lower-upper approximate factorization scheme for the solution of the system of linear equations. The boundary conditions employed in the present simulation are the combination of the relativistic boundary conditions and the characteristic variable (CV) boundary conditions. The former is to include the relativistic effects due to the movement of conductor and given by n E = (n v)b (4) where v and n are the instantaneous velocity of the oscillating conductor and the unit vector normal of the conductor surface, respectively. Each CV is defined as the product of one row eigenvector and the instantaneous variable vector. Because each CV is associated with one of the eigenvalues it carries information with the speed and direction that which information propagates. Since MOC evaluates flux at the cell face within cell, one of the characteristic variables carrying information and arriving on the boundary is given by CV b = n B + η o D (5) where η o is the impedance of free space and the superscript (b) is for the CV evaluated on the boundary, and variables B and D are taken from the adjacent cell. By using the fact that n =-1 as in the present problem, we solve for boundary values from (4) and (5) and have B b = 1 v 1 CV b (6) E b = v v 1 CV b. (7) The change in the field magnitude due to the relative motion between electromagnetic wave and perfect conductor is predictable by the multiplying factor 1 + β Max 1 β Max. (8) Above, β Max is the ratio of the maximum PEC instantaneous velocity to the speed of light and ranges from -0.1 to +0.1, respectively corresponding to the approaching and receding cases. The instantaneous velocity of the oscillating perfect conductor is designated to be negative if it moves toward the incidence and positive if it goes in the same direction as the incidence. Expression (8) is employed to investigate the accuracy of the present method and known as the Doppler shift. Cases Studied The problem is specified in Figure 1 where the incident wave is plane and initially propagates toward a vibrating perfect conductor. The incident electromagnetic wave is monochromatic (1 GHz in frequency or 30 cm in wavelength) and composed of seven complete cycles. A one-half Gaussian window is applied to each end to avoid abrupt changes in field quantities that are measured one full wavelength from the peak to the truncated

3 76 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August Figure 1: Definition of the studied problem. level (100 db). The vibration type of the vibrating perfect conductor is either zigzag or sinusoidal. The vibration frequency of the perfect conductor is set to be either 1 or 2 GHz. The corresponding peak-to-peak amplitudes are respectively and mm for sinusoidal and 15 mm and 7.5 mm for zigzag such that the maximum instantaneous velocity is one tenth of the light speed, i.e., β Max = ±0.1 near the equilibrium position. The calculated instantaneous velocities are given in Figure 2. It is clearly shown that the vibration forms are either Instaneous Velocity Figure 2: Calculated instantaneous velocities: β Max = ±0.1 (f i = 1 GHz). sinusoidal or periodic rectangular functions; one is involved with acceleration while the other is not. For a one-dimensional model the cell indexing is simple. Both the total number of grid cell and the cell size are constant values. In the present work, because the oscillatory behavior of perfect conductor as illustrated in Figure 3, portion of the N th cell may be occupied (left dashed lines) by the oscillating conductor surface (dark Figure 3: One-dimensional cell indexing with a vibrating boundary.

4 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August solid line, equilibrium position) at certain instant of time. A moment later an extra fractional cell, the (N+1) th cell, is added (right dashed lines). Results In order to observe how the oscillation type and frequency affect the reflected waves, the reflected electric fields are shown in Figure 4 where the vibration frequency ( ) of conductor is either identical with or twice Reflected Electric Field Magnitude Stationary Figure 4: Reflected electric field magnitudes (f i = 1 GHz). as high as that of the incident wave (f i ). It is shown that the variations of the reflected electric field bear the similar characteristics of the vibrating perfect conductor. The resultant magnitude and frequency of the reflected waves were calculated directly from the computation results. The Doppler shift values in the reflected electric field magnitude are also computed and compared with the theoretical values as listed in Table 1 where the exact values are obtained by using (8). As clearly seen they are in good agreement. It is noticed that when the perfect conductor is vibrating in zigzag the reflected electric fields overshoot in magnitude whenever the perfect vibrator changes direction rapidly at which time instance the perfect conductor experiences an abrupt change in velocity from to -0.1 and vice versa. Table 1: Comparison of the calculated with the theoretical values: magnitude and frequency of the reflected electric fields. Vibration Extreme Maximum Magnitude Minimum Magnitude Frequency (GHz) Type Velocity(C) Theoretical Calculated Theoretical Calculated Theoretical Calculated Sinusoidal ± Sinusoidal ± Zigzag ± Zigzag ± Given in Figure 5 are the phase differences for various cases in which they are calculated on the basis of the stationary case. The phase change reflects the oscillatory behavior of the vibrating perfect conductor in both amplitude and form. It is observed that the phase change depends on the direction of the moving perfect conductor: it changes more when the perfect conductor recedes from the incident than when it approaches. The pattern of the phase difference also reveals the vibration type: the change is linear when the perfect vibrator goes in zigzag and sinusoidal when it vibrates sinusoidally. The computational results give the fact that the object oscillation alters the reflected waves of a stationary object in phase as well as in magnitude and that these variations are subject to the type oibration.

5 78 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August Phase Difference( Reflected Electric Fields ) Stationary Degrees Figure 5: Phase difference in the reflected electric fields (f i = 1 GHz). Conclusion In this paper we present that the method of characteristics successfully simulates the effects oibrating perfect conductor on the reflected waves in one-dimension. The Doppler shifts in magnitude of the reflected electric fields are found in good agreement with the theoretical values. On the surface of the perfect conductor, we successfully applied the electromagnetic boundary conditions that are the combination of the characteristic variable boundary conditions and relativistic boundary conditions. The results showed that the reflected electric fields are imprinted with the properties of the oscillating perfect conductor. It is our future goal to extend the existing formulation to three-dimensional problems. REFERENCES 1. Whitfield, D. L. and J. M. Janus, Three-dimensional Unsteady Euler Equations Solution Using Flux Splitting, AIAA Paper No , Shang, J. S., A Characteristic-based Algorithm for Solving 3-d Time-domain Maxwell Equations, Electromagnetics, Vol. 10, 127, Donohoe, J. Patrick, J. H. Beggs and Mingtsu Ho, Comparison of Finite-difference Time-domain Results for Scattered EM Fields: Yee Algorithm vs. a Characteristic Based Algorithm, 27 th IEEE Southeastern Symposium on System Theory, Ho, M. M., Numerical Simulation of Induced Currents on a Slowly Moving PEC Plane under the Illumination of EM Pulses, Moving Boundaries 2003, Santa Fe, New Mexico, USA, Cooper, J., Scattering of Electromagnetic Fields by a Moving Boundary: the One-dimensional Case, IEEE Trans. Antennas Propagation, Vol. AP-28, No. 6, , Kleinman, K. E. and R. B. Mack, Scattering by Linearly Vibrating Objects, IEEE Trans. Antennas Propagation, Vol. AP-27, No. 3, , 1979.