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Chapter 1
Numerical Analysis Techniques
Ramesh Garg
Indian Institute of Technology, Kharagpur, India
1.1 Introduction
Microstrip and other printed antennas are constituted of, in general, patches, strips, slots, packaged semiconductor devices, radome, feed, etc. in a nonhomogeneous dielectric medium. Finite substrate and ground plane size are the norm. The dielectric used is very thin compared to the other dimensions of the antenna. The design of these antennas based on models such as transmission line model or cavity model is approximate. Besides, these designs fit regular-shaped geometries (rectangular, circular, etc.) only, whereas most of the useful antenna geometries are complex and do not conform to these restrictions [1]. The effect of surface waves, mutual coupling, finite ground plane size, anisotropic substrate, etc. is difficult to include in these types of design. The numerical techniques, on the other hand, can be used to analyze any complex antenna geometry including irregular shape, finite dielectric and ground plane size, anisotropic dielectric, radome, etc. The popular numerical techniques for antenna analysis include method of moments (MoM), finite element method (FEM), and finite difference time domain method (FDTD). MoM analysis technique, though efficient, is not versatile because of its dependence on Green's function. FEM and FDTD are the most suitable numerical analysis techniques for printed antennas. FDTD is found to be versatile because any embedded semiconductor device in the antenna can be included in the analysis at the device-field interaction level. This leads to an accurate analysis of active antennas. Maxwell's equations are solved as such in FDTD, without analytical pre-processing, unlike the other numerical techniques. Therefore, almost any antenna geometry can be analyzed. However, this technique is numerically intensive, and therefore require careful programming to reduce computation cost. We shall describe the advances in FDTD. Our reference in this respect is the classic book on FDTD by Taflove and Hagness [2].
A large number of FDTD algorithms have been developed. These can be classified as conditionally stable and unconditionally stable. The conditionally stable schemes include the original or Yee's FDTD also called FDTD (2,2), FDTD (2,4), sampling bi-orthogonal time-domain (SBTD) and their variants; and the unconditionally stable schemes include ADI (Alternate Direction Implicit), CN (Crank Nicolson), CNSS (Crank Nicolson Split Step), LOD (Local One-Dimensional) and their variants. The updating of fields in conditionally stable schemes does not require a solution of matrix equation as an intermediate step, and are therefore fully explicit. However, these schemes have a limit on the maximum value of the time step, which is governed by the minimum value of the space step through the Courant-Friedrich-Levy (CFL) condition.
Due to the heterogeneous nature of the dielectric in the printed antennas, the wave velocity is less than
c and may vary from cell to cell and from one frequency to another. We therefore introduce a safety margin and choose
uniformly to simplify coding and avoid instability. Defining the Courant number
q as
implies that
q = 1/2 and the wave takes
time to travel to the next node.
The value of
puts a severe computational constraint on the structures as they have fine geometrical features such as narrow strips or slots or thin dielectric sheets. Since the simulation time of an antenna is independent of space and time steps, the number of updates of fields increases linearly with the decrease in the time step. This results in an increase in processor time. The limitation on
is removed in some of the FDTD algorithms and these...
