Finite x infinite array

Consider next the finite x infinite array shown in Fig. 1.2. It consists, like the infinite x infinite case in Fig. 1.1, of columns that are infinite in the Z direction, however, there is only a finite number of these columns in the X direction. Such arrays have been investigated by numerous researchers [4-23]—in particular, by Usoff, who wrote the computer program SPLAT (Scattering fi om a Periodic Linear Array of Thin wire elements) as part of his doctoral dissertation in 1993 [24, 25]. 

Fig. 1.3 Various cases of a plane wave incident upon infinite as well as finite arrays at 45° from normal in the H plane. Element length 21=1.5 cm, load impedance Zl = 0 and frequencies as indicated, (a) Element currents for an infinite x infinite array at 10 GHz as obtained by the PMM program (close to resonance), (b) Element currents for a finite x infinite array of 25 columns at 10 GHz (close to resonance), (c) Element currents for a finite x infinite array of 25 columns at 7.8 GHz ( 25% below resonance).

Fig. 1.4 The bistatic scattered fieid in the H plane from a finite x infinite array of 25 columns...

For a. finite x infinite array as shown in Fig. 4.1, we will now show that the current will be ... 

Fig. D.2 A finite x infinite array comprised of 2Q+1 infinitely long stick arrays. All elements are driven with voltage generators and the currents in the reference elements in row 0 are denoted Iq. For Q -> oo we obtain an infinite x infinite array where Floquet s Theorem yields...

Fig, 4.26 Crude modeling of the faceted radome shown in Fig. 4.25. It is comprised of finite X infinite planar arrays. However, to avoid possible reflections from the back, only the four front panels are used to study the column currents in the front. 

This title will undoubtedly raise a few eyebrows. As stated in many respectable textbooks, surface waves do not radiate—period. What is not always emphasized is the fact that the theory for surface waves in general is based on a two-dimensional model like for example an infinitely long dielectric coated wire. And as discussed in this chapter infinite array theory may reveal many fundamental properties about arrays in general but there are phenomena that occur only when the array is finite. The fact is that surfaces waves are associated with element currents. They will radiate on a finite structure in the same manner an antenna radiates, namely by adding the fields from each column in an end-fire array. Numerous examples of this kind of radiation pattern will be shown in Chapter 4. They are typically characterized by having a mainbeam in the direction of the X axis that is lower than the sidelobe level. The reason for this abnormality is simply that the phase delay from column to column exceeds that of the Hansen-Woodyard condition by a considerable amount [29]. They also have a much lower radiation resistance. 

Fig. 4.4 Top If an infinite array is exposed to an incident plane wave propagating in tJie directions, S, it will reradiate propagating plane waves in the directions s = XSx 9 Sy + tSz and eventually a finite number of grating waves. In addition, evanescent waves are always present Middle A finite array with only Floquet currents will produce a contimous spectrum fc with a mainbeam and sidelobes as shown at the bottom. Note that we use the x component r of r as our variable. This is consistent with using s from the direction S of the incident plane wave. Bottom Far field from Floquet currents as a function of the continuous radiation direction fc.

A number of other parameters were investigated for. arrays of fuel assemblies. For example, reactivity of storage arrays is highly dependent on the finite size of the array. Keff for a finite array of 100 assemblies on a 21- X 38-in. pitch is <0.9 at 5% interspersed moderation versus 1.10 for the infinite array. 

Fig. 1.2 An array that has a finite number of eiement coiumns in the X direction and is infinite in the Z direction, it is tmiy periodic in the iatter direction but not in the former. Thus, Fioquet s Theorem appiies oniy to the Z direction, not the X direction.

Finally we recall that the SPLAT program is structured to have the finite dimension along the x axis and the infinite along the z axis. Thus, the array... 

The components of the vectors X and v denote the coordinates of any two fuel rods in the finite array, as located from some arbitrary reference point. The result (10.261) is a system of homogeneous equations for the thermal fluxes at the surfaces of the fuel rods 0Af(X), A solution to this system exists if the determinant is identically zero. By imposing this requirement the determinant reduces to an algebraic equation, and the roots of this equation yield the condition for criticality. As in the case of the infinite lattice, all four of the basic parameters will be involved in this equation, and by specifying any three, the determinant will yield the fourth. The calculational procedure involved here is demonstrated in a subsequent analysis, and an example of a four-rod reactor is computed in detail. 

---