Stick arrays

More specifically, we shall investigate arrays comprised of infinitely long columns along the z axis, where the elements have arbitrary orientation p = Px +yPy +zPz- Such single-column arrays are often called stick arrays. They are simply the building blocks for more complicated arrays and are therefore extremely important to investigate. 

Fig. 3.1 Infinite stick array of Hertzian dipoles with arbitrary orientation p=Xpx - -9Py + Pz and eiement currents = io The reference eiement is located at (0, 0, z ).

Finally, the mutual impedance between a stick array with an external element is defined as... 

When all the mutual and the self-impedances of an assembly of stick arrays as shown in Fig. 3.8 has been determined, it is a relatively simple matter to find the currents on the elements. In fact, it was already discussed in my earlier book [62] and need therefore only to be restated here for easy reference. 

Let the stick arrays in Fig. 3.8 be exposed to an incident plane wave with direction of propagation equal to s. The voltages induced in the reference element of the three or more stick arrays are denoted y(2) yo) respectively. They are easily determined by application of (3.18). Similarly, the currents on each reference element are denoted by I and respectively. 

Fig. 3.8 An arbitrary but finite number of infinitely long stick arrays with the same interelement spacing Dz is exposed to an incident plane wave E,. Element orientation and length may vary from stick array to stick array.

In this chapter we have obtained the E field from stick arrays composed of longitudinal or transverse elements of arbitrary length 21. 

Consider a stick array with longitudinal elements as shown in Fig. 3.2 with element lengths 2Z = 1.5 cm and = 1.6 cm. The incident field is arriving at broadside with s = y and 1 =zE Furthermore, the wire radius a is... 

Compare the convergence properties for the stick array in Problem 3.1 with that of the infinite x infinite array in Problem 3.2. 

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... 

Fig. 4.1 The array considered here consists of a finite number of infinitely long columns (stick arrays) with axial elements. Voltages are induced in all elements by an incident plane wave propagating in the direction S = kSx+ Sy + Is. We seek the bistatic scattered pattern.

Periodic structures of finite extent are often analyzed by dividing them into infinitely long column arrays (stick arrays) with either longitudinal or transverse elements. Thus, in Chapter 3 we examined both of these types of arrays in some detail. 

Consider Fig. D.l, where we show two infinite stick arrays q and q. We denote the mutual impedance between the reference element in array q and element m in array q by The mutual impedance between the reference element in... 

Let us now consider a finite x infinite array as shown in Fig. D.2 consisting of 2Q + stick arrays. The current in the reference elements are denoted Iq. Writing Ohm s Law for the reference element in stick array number 0 yields... 

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...

The embedded stick impedance is perhaps best illustrated by considering just three stick arrays as shown in Fig. D.4. Each element in the center column is fed by voltage generators with voltages V° ", where m refers to the row number. The two outer stick arrays are not fed but only loaded with identical load impedances Zl. 

Fig. D.4 The embedded stick impedance Zemb stk is the terminal impedance obsen/ed in the center stick array when all terminals in the center array are fed with voltages and all other stick arrays are loaded with identical load impedances and merely excited parasitically.

It can be shown that the embedded stick impedance for any size finite array is structured the same way—that is, as a sum of the stick self-impedance Z° ° plus overcoupled terms associated with all the other parasitically excited stick arrays. 

It is therefore of interest to investigate just a single stick array when we feed only a single pair of terminals while the rest are loaded with the same load impedances Z. We have denoted the terminal impedance for this case for the embedded element stick impedance Zemb eie stk- Examples are shown in Fig. D.7. The array has the same dimensions as nsed in the previons section (see insert). The calcnlations were obtained from the method of moment program ESP [137]. Similar to the SPLAT program nsed to obtain the resnlts in Figs. D.5 and D.6, it uses dielecttic cylinders placed aronnd each element. 

Fig. D.6 The embedded stick impedance Zemb stk as given by. 10) (no groundpiane). The center stick array is driven whiie the two outer stick arrays are Just haded with Zl = 100 ohms. Array dimensions as in Figs. D.3 and D.4 (see insert). From the SPLAT program, inciudes matching transformer (see text).

An even greater difference is observed if we consider the embedded element stick impedance Zemb eie stk as shown in Fig. D.7. Here we excite only the center element of a single stick while all the other elements in the stick array are parasitically excited. 

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