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Unformatted text preview: 1 EE256 Numerical Electromagnetics H. O. #6 Marshall 24 June 2008 Summer 2008 LECTURE 5 5.1 FDTD EXPRESSIONS IN THREE DIMENSIONS The FDTD expressions for Maxwells Equations in three dimensions are given in Section 3.6.3 of the text- book, Computational Electrodynamics , by Taflove and Hagness. However, noting that the placement of the electric and magnetic fields on the FDTD cell is somewhat different between the Lecture Notes and the text, we provide expressions for one electric field and one magnetic field component. y y x z x x y z y z x x z y x y ( i,j,k ) ( i-1 ,j +1 ,k) ( i-1 ,j +1 ,k +1) ( i,j +1 ,k ) ( i,j,k +1) z z Figure 5.1: Placement of electric and magnetic field components in a three-dimensional staggered mesh, known as the Yee cell. H z n +1 / 2 i +1 / 2 ,j +1 / 2 ,k = H z n- 1 / 2 i +1 / 2 ,j +1 / 2 ,k [5.1 a ] + t i +1 / 2 ,j +1 / 2 ,k E x n i +1 / 2 ,j +1 ,k-E x n i +1 / 2 ,j,k y- E y n i +1 ,j +1 / 2 ,k-E y n i,j +1 / 2 ,k x E z n +1 i,j,k +1 / 2 = E z n i,j,k +1 / 2 [5.1 b ] + t i,j,k +1 / 2 H y n +1 / 2 i +1 / 2 ,j,k +1 / 2-H y n +1 / 2 i- 1 / 2 ,j,k +1 / 2 x- H x n +1 / 2 i,j +1 / 2 ,k +1 / 2-H x n +1 / 2 i,j- 1 / 2 ,k +1 / 2 y Note that the above expressions are for the case where we have no electric or magnetic current sources, J i or M i . Inclusion of the sources is straightforward, as shown in Section 3.6.3 of the textbook. 2 5.2 FDTD ALGORITHM FOR LOSSY MEDIA In the previous lecture, we derived the 1-D and 2-D FDTD algorithms by neglecting the loss terms in the two curl equations. Incorporation of these losses into the FDTD algorithm is straightforward, as discussed below. We start with the 1-D case, which we pose in terms of uniform transmission line equations, noting that the algorithms we derive for...
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