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ods are probably the most commonly used methods for analysis of antennas, including conformal arrays This class is often subdivided into low-frequency methods and highfrequency methods High frequency implies that the properties and size parameters of the geometry vary slowly with the frequency In this chapter, we will use a somewhat different classification of the methods: electrically small surfaces and electrically large surfaces Both frequency and time domain methods are included in a logical way, making it easier to find a suitable method for a practical application The distinction between electrically small and electrically large surfaces is not always clear the two categories overlap to some extent The assumption of an electrically large surface is often valid for large bodies when the frequency is in the microwave frequency band or higher However, it is necessary to know several methods and find out when they are applicable.

utilization of methane, for example, in oxidative coupling 416 A Ga loading as low as 05% was sufficient to obtain maximum yields of aromatic products At 520 C, an ethylene conversion of 93%, with an aromatics selectivity of 81 % was obtained over a 5% Ga-ZSM-5 catalyst The study also showed that Ga exists as both Ga3+ at zeolitic exchange sites and as Ga203 within the channels and on the external surface of the calcined catalyst It is also important to point out, that aromatization over all Ga-Ioaded zeolite catalysts strongly influenced by operating conditions Specifically, product selectivity in the aromatization of propane depends on conversion,417-419 that is, space velocity and temperature 419 New high-surface-area zirconia-carbon composites were prepared by a sol-gel method followed by a high-temperature treatment in inert gas.

A hybrid approach (a combination of different methods) is often used in practice We will focus on the most commonly used methods but not describe the methods in great detail Instead, we will try to point out what to take into account and describe the features of the methods, both good and bad With this information, we hope the reader will be able to find a suitable method when facing a conformal antenna problem However, the practical application of a certain method is not discussed; this is up to the user to find in the relevant references Some help is provided by the many references that are listed, and also by two canonical examples that will be described in Section 45 Before entering into the discussion of the various methods we present in Section 4.

TABLE 33-3

2 the typical electromagnetic problem that we have to solve, and we introduce also some parameters that will be used in the next several chapters Section 43 on electrically small surfaces deals with frequency domain methods such as modal solutions, integral equations (IE)/method of moments (MoM), and the finite element method (FEM) Time domain methods are also included here, including the most commonly used, the finite-difference time-domain method (FDTD) These methods become intractable for structures that are large in terms of the wavelength, since the associated matrices become very large Thus, an upper frequency limit exists for these methods The time domain methods are often used for analyzing antennas for large frequency bands, which, however, requires much computer memory (and time) The classical FDTD has a disadvantage when applied to conformal antennas: the staircase approximation of curved surfaces Some attempts to improve the method will be discussed Section 4.

4 discusses methods for electrically large surfaces The methods considered here have a lower frequency limit Examples of methods suitable for this class of surfaces are geometrical optics (GO), geometrical theory of diffraction (GTD), uniform theory of diffraction (UTD), physical optics (PO), and physical theory of diffraction (PTD) In Section 45, we present some more details for two canonical problems The two cases are a waveguide-fed aperture antenna and a microstrip-patch antenna located at an arbitrarily shaped surface The approach is an integral equation formulation, solved with the method of moments The reason for choosing this approach is that we can point out several interesting aspects, and at the same time retain a physical insight into the solution With a full numerical approach, this insight is more difficult to illustrate Section 46 concludes this chapter with a short discussion and comparison of the various methods.