Finite Array

8 ON THE RADIATION FROM INFINITE VERSUS FINITE ARRAYS 2.8.1 Infinite Arrays 

It is well known that the field radiated from an infinite array can be written as a sum of a finite number of propagating waves (the principal term corresponding to n = 0, 0 plus grating waves if any) and an infinite number of evanescent waves [50]. An example is shown in Fig. 2.11, top, where the direction of the principal propagating wave is denoted r(0, 0) and where no grating waves are encountered. Also indicated are the evanescent waves. They will typically die out as the point of observation moves beyond a distance A/4 from the array at which point only the propagating waves will exist. Thus, when dealing with infinite arrays we will always be in the near field as far as the radiation pattern is concerned the concept far field simply does not exist for an infinite array. 

The field radiated from a finite array is shown quantitatively in Fig. 2.11, bottom. Close to the array the field looks approximately as for the infinite array except for some ripples. These are merely caused by the same type of surface waves encountered in Chapter 1 and discussed in more detail in Chapters 4 and 5. They typically occur every time a general aperture is finite rather than infinite. 

As we move away from the finite array to a distance about half the size of the aperture we observe that the extent of the field becomes narrower than the aperture. And at a distance about the size of the aperture it has an extent about the size of the aperture. These observations are typical. 

Finally, we show a polar plot of the far field obtained by simple aperture integration. Comparing the infinite and finite cases we observe that the former has an infinite narrow beam (like a Dirac-Delta function) in the direction r (0,0) while the latter exhibits the well known far field pattern. At the array elements they both look similar except that the finite case shows ripples along the aperture as caused by the presence of surface waves as discussed in Chapters 1, 4 and 5. 

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A finite array of charges is built taking into account the symmetry elements of the crystal. The charges of the outermost ions are adjusted in order to provide the correct value of the Madelung potential on each cluster site as well as the electrical neutrality... 

Dealing with boundaries becomes an issue in wavelet analysis of finite array of data. These edge effects or singularities can be avoided by using adaptive filters near the signal boundaries. 

Problems with the Discrete, Finite Array Fourier Transform... 

The assessment might involve the calculation of large finite arrays for which there is a lack of experimental data. Therefore a specific supplementary allowance should be made in addition to other margins usually allowed for random and systematic effects on calculated values of the neutron multiphcation factor. 

VII.56. The objective of the package array calculations is to obtain the information needed to determine the CSI for criticality control as prescribed in para. 528. The assessor may consider beginning the array calculations with an infinite array model. Successively smaller finite arrays may be required until the array sizes for normal and accident conditions of transport are found to be below the USL. As an alternative, an applicant may initiate the analyses using any array size — for example, one that is based on the number of packages planned to be shipped on a vehicle. 

The data contained in a digitally recorded image is an ordered finite array of discrete values of intensity (grayscale). To manipulate this data, the continuous integrals defining the Fourier transform and convolution must be expressed as approximating summations. For a series of discrete samples jc( t) of a continuous function x(t), a representation in the frequency domain can be written as... 

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).

We observe in Fig. 1.3c that the element currents for the finite array not only fluctuate dramatically from column to column but also exhibit an average current that can be estimated to be somewhat higher than the currents even for resonance condition (0.055 mA). 

The so-called end currents. These are prevalent close to the edges of the finite array and are usually interpreted as reflections of the two surface waves as they arrive at the edges. 

We emphasize that these surface waves are unique for finite arrays. They will not appear on an infinite array and will consequently not be printed out by, for example, the PMM program that deals strictly with infinite arrays. Nor should they be confused with what is sometimes referred to as edge waves [28], The propagation constant of these equals that of free space, and they die out as you move away from the edges. See also Section 1.5.3. 

Furthermore, the surface waves here are not related to the well-known surface waves that can exist on infinite arrays in a stratified medium next to the elements. These will readily show up in PMM calculations. These are simply grating lobes trapped in the stratified medium and will consequently show up only at higher frequencies, typically above resonance but not necessarily so in a poorly designed array. In contrast, the surface waves associated with finite arrays will typically show up below resonance (20-30%) and only if the interelement spacing is <0.5)t. 

Fig. 1.5 The actual element cunents in each column of a finite array of 25 columns when exposed to an incident plane wave at 45° from normal or fed like a phased array from individual voltage generators with a linear phase delay, (a) All elements loaded with Ri = 100 ohms, (b) Only the outer columns (each side) are loaded with 200 ohms, the next inner columns with 100 ohms and finally the third inner ones with 50 ohms. All other elements have no load resistances, (c) All elements baded with Rl = 20 ohms.

An alternative approach is to assume that the radiation from a finite array is associated entirely with the edge currents. While Maxwell s equations do not state specifically that radiation or scattering takes place from neither edges or element tips, it is nevertheless an observation that has proven valuable in classical electromagnetic theory. It is a convenient way to handle scattering properties from perfectly conducting half-planes, strips, wedges, and more, even when made of dielectric. 

However, in the case of finite arrays of loaded wire elements the approach loses some of its appeal by the fact that snrface waves exist only in a limited frequency range inside which the amplitude and phase vary considerably with frequency. Consequently, the scattering properties must be calculated numerically at each frequency and will actually also depend on array size in a somewhat complicated way. 

As shown in Chapter 4, each of the two surface waves are generated from semi-infinite arrays located adjacent to the finite array. We will assume the equivalent circuit to consist of surface wave generators at each end of the finite array with surface wave generator impedances for the left- and rightgoing surface wave denoted Zsw l and Z w r, respectively. We will assume that these impedances depend on angle of incidence. 

Fig. 2.11 Top The field from an infinite array cxinsists of a finite number of propagating plane i/i/ai/es and an infinite number of evanescent waves. The sum of the evanescent waves essentially constitutes the near field of the array. Bottom The field from a finite array at various distances as shown.

ON TRANSMITTING, RECEIVING, AND SCATTERING RADIATION PATTERN OF FINITE ARRAYS... 

In the last section we introduced the radiation pattern concept for finite arrays. We shall in this section consider in more detail what happens under transmitting versus receiving or scattering conditions. 

Consider a finite array being fed from a single pair of terminals via some kind of harness. If we transmit from those terminals it is a conceptually simple process to obtain a far field transmit pattern Paf based on the element currents under transmitting condition. 

In Fig. 2.12, middle, we show a large finite array of dipoles backed by a finite groundplane as large as the array. When fed from constant current generators, we obtain the currents under transmitting condition, and the transmit pattern Paf is obtained in the traditional way based on these currents and their approximate images in the finite groundplane. 

Final Remarks Concerning Transmitting, Receiving, and Scattering Radiation Pattern of Finite Arrays... 

It was primarily for that reason that Usoff decided to work in the spatial domain when he wrote the SPLAT program. He further used a shanks transform to obtain faster convergence [24]. It became a most wonderful and versatile program. In fact, it is the workhorse for our research into finite arrays. 

Most of these expressions were not available to Usoff when he was given the task to write a program for finite arrays. He therefore chose to stay mostly in the spatial domain except for the longitudinal case. The result was a program that was extremely versatile and became the workhorse for our finite array research. It was later given a face lift by Dan Janning and other students that reduced the run time and made it more user friendly. 

It is doubtful whether any of the author s students will ever attempt to write a program in the spectral domain for finite arrays like originally envisioned (I am running out of time, students, and money ). But if anyone out there does, please let me hear about it. 

The finite array will be modeled by a finite number of infinitely long column arrays (also called stick arrays see Fig. 4.1). This approach has been widely used by several researchers [74-80]. One of them, Usoff, wrote as part of his dissertation [24] the computer program Scattering fi om a Periodic Array of Thin Wire Elements (SPLAT). The excitation can be either in the form of an incident plane wave propagating in the direction 5 = + ysy + zsz (passive case). Or... 

When analyzing the finite array, it is advisable first to review the infinite array case that is, the finite number of columns shown in Fig. 4.1 becomes infinite as shown earlier in Fig. 1.1. As discussed for example in reference 62 or Chapter 3, infinite arrays are significantly simpler to analyze than finite arrays. In particular, if exposed to an incident plane wave with direction of propagation equal to s, the element cnrrents are all related to each other by Floquet s Theorem ... 

Fig. 4.2 Typical scan impedance for an infinite and a finite array at the fixed frequency 8 GHz (i.e., below resonance) as a function of scan angle. Note By feeding the elements with actual voltage generators like a phased array, we can scan the beam" beyond endfire into the imaginary space where Is l > 7 and only evanescent waves are possible, provided that the interelement spacing Dx is < 0.5 k.

THE ELEMENT CURRENTS ON / FINITE ARRAY EXCITED BY AN INCIDENT WAVE... 

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