Applications of Time Domain Vector Potential Formulation to 3-D Electromagnetic Problems

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3 Applications of Time Domain Vector Potential Formulation to 3-D Electromagnetic Problems F. De Flaviis, M. G. Noro, R. E. Diaz, G. Franceschetti and N. G. Alexopoulos Department of Electrical Engineering Department of Chemistry and Biochemistry University of California at Los Angeles, Los Angeles, CA Abstract -- The introduction of the scalar and vector potentials allows to rewrite Maxwell s equations with a different formalism. Upon discretization, we exploit the advantages of the condensed node representation of the fields, resulting in lower grid dispersion, memory and computational time saving. Numerical examples are presented to validate the technique in two and three dimensions. I. INTRODUCTION A number of numerical techniques have been developed in the past for the solution of electromagnetic problems in the time domain [1]. Much work has also been devoted to the formulation of condensed node discretizations where the components of the field are located at the same point in space on the computational grid [2][3]. The use of our vector potential formulation (VP) offers the advantage a condensed node representation of all the quantities involved in the simulation [4][5]. It also differs from previous vector potential approaches in that the electric field is used as one of the time-evolving quantities. The condensed node representation of the field results in lower grid dispersion compared to other numerical techniques and leads to a memory and computational time saving up to 38%. II. VECTOR POTENTIAL FORMULATION Upon introduction of the vector potential it is possible to recast Maxwell's equations in the following form Ï Ì Ó m ( e E) 2 = ( A) - A - m H - sme A = -E - f A = -em f (1) were A and f are, respectively, the electric vector potential and the scalar potential. Previous work has shown that one and two dimensional expressions of Eq. (1) can be used to solve scattering problems from electrically lossy and permeable materials [6]. Further extensions of the model lead to the treatment of frequency dependent materials [7][8]. In this paper we will exploit the vector potential formulation to solve two and three dimensional problems comparing results with traditional Finite Difference Time Domain (FDTD) [1] and analytical solution, where available. III. RESONANT CAVITY We present in this section a typical eigenvalue problem, and compare the numerical results obtained with this technique with the theoretical predicted values. A cubic resonator is excited with an electric field pulse, and, after steady state is reached, by means of Discrete Fourier Transform (DFT) [9] the resonant frequencies are extracted. These are compared with the exact theoretical values for the m,n,p mode calculated from [10] : f c Ê np ˆ Ê np ˆ Ê pp ˆ,, = Á + + Ë L Á Á 2p Ë L Ë L mnp x y z where L x, L y and L z are the linear dimensions of the resonator and c is the speed of light. In our experiment we use a cubic resonator so all the dimension are identical. The space was discretized by using 23 cells along each side; the dimension of each cell is 0.2 mm. A Gaussian pulse in shape is applied in the geometrical center of the resonator; the width of the pulse is 40 time steps corresponding to 20 psec. The simulation is time stepped for a long enough time so that a steady state is reached, typically 8,000 steps. DFT algorithm is used at a non zero field point, away from the center and from the boundary to obtain the frequency resonances of the resonator. The DFT analysis of the z-component of the electric field was performed using 1000 points to represent a span of 120 GHz centered around 70 GHz to localize all the excited modes. The resonance frequencies are shown in Fig. (1). (2)

4 Magnitude Fig.1 Magnitude of E z field component plotted versus frequency calculated in the 3-D simulation Notice that the amplitude of each peak depends on the particular choice of the observation point, care must be used to avoid null points. The frequency of the fundamental mode is determined experimentally as the position of the first maximum in Fig.(1), and it coincides within an error less than 1% with the analytical result calculated from eq. (2). The same analysis was repeated for every other mode and the error was always of the same order. IV. COMPARISON WITH OTHER NUMERICAL TECHNIQUES The key feature of the technique presented in this work resides on the condensed node representation of the field components. Once space and time are discretized, in order to solve Maxwell's Equations in the vector potential (VP) formulation, all the components of the quantities involved in the model refer to the same location of the computational grid. This property is common also to other established numerical techniques such as condensed Transmission Line Matrix (TLM)[4][5], while differs substantially from the well known Yee representation scheme exploited by FDTD[11]. In Fig. (2) we plot the field components used in the two dimensional FDTD cell (Ez, Hx, Hy) and the analogous components used in our vector potential formulation (Ez, Az). Note that the FDTD components are dislocated along the sides of the cell, while the vector potential components are both located in the center of the cell. x Hx z FDTD Ez 52.96GHz Hy y 83.44GHz Frequency (Hz) Fig.2 Comparison of the field components location in the elementary cell between 2-D FDTD and VP As a consequence of the fact that the field is condensed, our technique offers superior performance in terms of grid dispersion with respect to x 85.12GHz z VP GHz GHz Ez Az y the FDTD formulation, as it was already proven in the literature for the analogous TLM. At the same time no price has to be paid in terms of memory requirements for all the cases considered, and this is an advantage with respect to TLM. Furthermore for the two dimensional case considered below only two quantities need to be considered and stored in time (Ez, Az) as opposed to the FDTD representation where three components need to be used (Ez, Hx, Hy). This results in one third memory saving over FDTD, with no penalty in execution time and algorithm complexity. In the following numerical experiment we evaluate the cutoff frequency due to grid dispersion in a computer experiment and we compare it with the same quantity evaluated in the analogous FDTD experiment. A locally plane wave tilted with respect to the grid main axes is generated as shown in Fig (3). 0.16m 0.1m 0.16m observation point Fig.3 Field distribution and experimental geometry for the grid dispersion numerical experiment The wave is generated by a Gaussian pulse 62.5 fsec in width corresponding to a maximum frequency content of 200 GHz. The wave front extends for 100 cells across corresponding to 2.0 cm, and the observation point is placed along the perpendicular to the wave front, 7.07 mm away from the wave source location. This geometry was chosen so that the wave is almost plane when it leaves the observation point. The mesh size is coarse enough such that grid dispersion will occur. As shown in reference [12], for a wave propagating at 45 along the FDTD grid, the grid dispersion causes the wave propagation velocity to fall to zero when Ds>0.5l, where l is the wavelength of the electromagnetic wave examined. The corresponding cutoff frequency for our particular choice of Ds is fmax=150ghz. The simulation is time-stepped for a long enough time so that the Gaussian pulse goes entirely past the observation point. The time responses are recorded in both cases. Due to grid dispersion, the original pulse is altered and develops a ringing tail. DFT analysis gives the spectrum content which is shown in Fig. (4), where we plot the magnitude of the two spectra. Notice the sudden drop of the magnitude at the cutoff frequency.

5 Magnitude E(f) Frequency (Hz) Fig.4 Spectrum content of the signal observed at the same observation point for VP (solid line) and FDTD (broken line). The observed FDTD cutoff frequency agrees with the predicted value of 150GHz, while our technique exhibits the cutoff at 170GHz, corresponding to a 13% improvement in bandwidth. The same bandwidth (170GHz) can be achieved by FDTD if the mesh size is reduced to Ds= m corresponding to a memory increase of 38% and execution time increase of 38%. V. CONCLUSIONS A new approach for the solution of electromagnetics problems has been proposed. The scheme takes advantage of the simplicity of the leap-frog scheme between the electric field and the vector potential, but maintains the higher performances of TLM condensed node due the nature of the discretization. The condensed node character of this time domain formulation results in lower grid dispersion with respect to FDTD. Our analysis concludes that, in order to obtain the same bandwidth, an increase in 38% in memory size and an analogous increase in computational time, is required for the FDTD two dimensional case. [5] F. De Flaviis, M. Noro, R. E. Diaz and N. G. Alexopoulos, The Diaz-Fitzgerald Time Domain Model for the Solution of Electromagnetic Problems IEEE AP-S Int. Symposium Montreal Canada, (1997). [6] F. De Flaviis, M. Noro, R. E. Diaz and N. G. Alexopoulos Diaz- Fitzgerald Time Domain (D-FTD) Technique Applied to Electromagnetic Problems IEEE MTT-S Int. Microwave Symp. San Francisco (1996). [7]F. De Flaviis, M. Noro, R. E. Diaz, G. Franceschetti and N. G. Alexopoulos Extensions to Complex Materials of the Fitzgerald Model for the Solution of Electromagnetic Problems submitted to Journal of Computational Physics [8] F. De Flaviis, M. Noro, R. E. Diaz and N. G. Alexopoulos Diaz- Fitzgerald Time Domain Method Applied to Electric and Magnetic Debye Materials Applied Computational Electromagnetics, ACES Symposium. Monterey California (1997). [9] L. Lapidus and G. F. Pinder, Numerical Solutions of Partial Differential Equations in Science and Engineering. (John Wiley and Sons, New York, 1982). [10] C. A. Balanis, Advanced Engineering Electromagnetics. (John Wiley and Sons Inc., New York, 1989). [11] K. S. Yee, Numerical Solutions of Initial Boundary Value Problems Involving Maxwell's Equations in Isotropic Media, IEEE Trans. Antennas Propagat. AP-14, 302 (1966). [12] A.Taflove, Computational Electromagnetics - The Finite Difference Time Domain Method, (Autech House Inc., Boston, 1995). REFERENCES [1] K. Luebbers, The Finite Difference Time Domain Method for Electromagnetics. (CRC Press, Boca Raton, Florida, 1993). Microwave Theory Tech. MTT-38, 919 (1990). [2] P.B. Johns, The Solution of Inhomogeneous Waveguide Problems Using a Transmission-Line Matrix, IEEE Transaction on Microwave Theory and Tech., MTT-22, 209 (1974) [3] W.J.R. Hoefer, The Transmission Line Matrix (TLM) Method in Numerical Techniques for Microwave and Millimiter-wave Passive Structure, Edited by Tatsuo Itoh, (Johns Wiley & Sons Inc., New York, 1989). [4] R. E. Diaz, A Discrete FitzGerald Time Domain Method for Computational Electromagnetics, in International Conference on Electromagnetics in Aerospace Advanced Applications (ICEAA). Politecnico di Torino, ITALY 1993, pp