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Unformatted text preview: 1 EE256 Numerical Electromagnetics H. O. #13 Marshall 3 July 2008 Summer 2008 LECTURE 8 Our subject for this Lecture is by and large well covered in Chapter 5 of the textbook, Taflove and Hagness, Computational Electrodynamics . We thus provide only abbreviated notes and expect the students to read the text in detail. 8.1 INTRODUCTION OF INTERNAL SOURCES A given FDTD formulation can be driven by either external (i.e., outside the FDTD grid) or internal (i.e., within the FDTD grid) sources. A common method for introduction of internal sources is to use the socalled hard source , which is setup by simply assigning a desired temporal variation to specific electric or magnetic field components at a single or a few grid points. Examples of hard sources for one dimensional TE modes are the continuous sinusoid, the Gaussian pulse, and the modulated Gaussian pulse given by E y n i s = E sin(2 πf n Δ t ) E y n i s = E e ( n 3 n τ ) 2 /n 2 τ E y n i s = E e ( n 3 n τ ) 2 /n 2 τ sin[2 πf ( n 3 n τ )Δ t ] where n τ Δ t is the characteristic halfwidth (in time) of the Gaussian pulse. Note that choosing the total duration of the Gaussian to be 6 n τ ensures a smooth transition both from and back to zero. More generally, for a two or three dimensional system, an internal source can be specified as a driven current, for example as a current density J source ( r ,t ) which is specified in terms of temporal form and at one or more grid points. This source current works to generate the electromagnetic fields via the curl equation: ∂ E ∂t n +1 / 2 = 1 [ ∇ × H ] n +1 / 2 1 [ J source ] n +1 / 2 where the source current is located at the same spatial grid location as the electric field but at the same time point as the magnetic field. Another simple source is a driven voltage applied across a gap, such as at the terminals of a wire antenna. Assuming a voltage source applied over a onecell wide gap in the y direction, we can write: E y n i,j = V ( t ) Δ y = V ( n Δ t ) Δ y 2 Once again, any given temporal variation can be imposed, by simply specifying the functional form of V ( t ) . Note that the antenna wire itself may extend over a number of mesh points ( i,j ± k ) on both sides of ( i,j ) , and its presence is simply incorporated into the FDTD calculation by forcing all E y values to be zero at these mesh points. 8.2 TOTAL AND SCATTERED FIELDS Our ability to separate the fields into incident and scattered components derives from the linearity of Maxwell’s equations and the fact that each of these components separately satisfy the two curl equations which are modeled in an FDTD algorithm. In general terms we can think of a free space region within which an incident field is specified analyt ically, being defined as the field which is present in this region in the absence of any objects (dielectric or conductor) or ‘scatterers’. The incident field thus satisfies Maxwell’s equations for free space, everywhere in the regions being modeled. Thein the regions being modeled....
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 '09
 Electromagnet, ez, HY, Incident plane wave, FDTD, field components

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