Scan impedance

For further developments to obtain spatial resolution such as scanning impedance spectroscopy 299 the reader is referred to the current literature. 

Variation of the Scan Impedance from Column to Column... 

Obviously it would be too much of a challenge to match an impedance with precision to the fluctuating scan impedance of Fig. 1.3c—in particular, when we realize that the maximum and minimum will start moving around with scan angle and frequency. Thus, we must simply look for ways to get rid of the surface waves or at least reduce them. We will discuss these matters next and in more detail in Chapter 4. 

Here the voltage generator is connected in series with its generator impedance Zq and the scan impedance Z. The ratio between the currents... 

This observation is quite noteworthy. It shows that by matching an antenna in the neighborhood of maximum power transfer (i.e., conjugate matching) we obtain an added benefit, namely a potential strong reduction of the ripples of the scan impedance even at a frequency where the surface waves are dominating. 

If the strucmre is used as a phased array, it can lead to dramatic variations of the terminal or scan impedance from column to column. Under these circumstances it would be very difficult to design a high-quality matching network in particular since the maximas and minimas of the scan impedance will move significantly with frequency and scan angle. 

Consider a phased array with scan impedance Za = 200 ohms. It is being fed from a generator with impedance Zc as shown in Fig. 1.6a. Assume conjugate match—that is, Zq = Za= 200 ohms. 

If used as a phased array, surface waves can lead to a significant variation in scan impedance from element to element, making precise matching difficult (see Figs. 1.3 and 1.5). 

A typical idealized example of the scan impedance for an infinite array with interelement spacing D /k 0.24 at / = 8 GHz is shown in Fig. 4.2. The total length of the elements is 2/ = 1.5 cm that is, the array will resonate around 10 GHz. Thus, we observe that the scan impedance at / = 8 GHz will be located in the capacitive part of the complex plane as also seen in Fig. 4.2. 

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.

Because of the lossy component of the scan impedance, the waves on the finite array can be considered as being surface waves for a slightly lossy periodic structure. 

Fig. 4.5 (a) An infinite an-ay exposed to an incident piane wave has oniy Fioquet currents, (b) Adding two semi-infinite arrays with negative Fioquet currents creates a finite array with actuai currents (Fioquet and residuat currents), (c) Spectrum of the voitages induced in the finite array by the incident wave, (d) Spectrum of the voitages in the finite array by the two semi-infinite arrays. Note the two peaks in the endfire directions rex = 1.(e) The magnitude of the scan impedance as a function of rex- Note the minima at rex = 125. (0 The spectrum of the eiement currents as a function of rex. Note the surface waves where Za is minimum is at rex = 1.25, not at rex = 1-0. 

Similarly, the voltage spectrum from the two semi-infinite arrays is shown in Fig. 4.5d. We observe two smaller beams at = 1, that is, in the endfire directions. Thus, the two semi-infinite arrays will try to propagate waves along the array structure. However, as illustrated in Fig. 4.2, many evanescent waves with different Tcxisx) are capable of propagating. They are distinguished by the magnitude of their scan impedance depicted in Fig. 4.2 and shown specifically in Fig. 4.5e. 

But how do we explain that surface waves are present only in a limited frequency range, namely 6.3-8.3GHz This is explained in Fig. 4.16 in a qualitative way. It shows the scan impedance plotted earher in Fig. 4.2 but here plotted at three frequencies, namely 6, 7.7, and 10 GHz. Furthermore, these scan impedance curves are shown a bit more realistic by the fact that they for end-fire condition do no go to infinity but just to a large value depending on how large the array actually is. This point is easy to see by application of the mutual impedance concept. It simply tells that the magnitude Za of the scan impedance can never exceed Ylq=-Q Zo, , where Zo,q is the mutual impedance between the reference element in column 0 and all the elements in column q (see Chapter 3 for details). Since Zo, j and Q are bounded, so is the finite sum Za. As already shown in Fig. 4.2 and repeated in Fig. 4.16 for easy comparison, we... 

Fig. 4.16 Typical complex scan impedance Za for a finite array at the following frequencies lOGHz, no surface waves since lowest point for Sx = 1.66 is too far from (0,0) 7.7GHz, surface wave possible for s 7.25 6.0 GHz, no surface wave since too far from (0,0).

We shall investigate the F-plane case analogous to the H-plane case above, namely by plotting the scan impedance from broadside (Sx = 0) and aU the way into the end of imaginary space and back into real space. 

A typical example of an F-plane scan impedance for an infinite array is shown in Fig. 4.33. We start at broadside at rj = 0° for = 0 and proceed to grazing at T] = 90° for = 1, which marks our entrance into the imaginary space. As Sx becomes larger than 1, we see that the scan impedance moves downward along the imaginary axis until it reaches its lowest level for = 112 from where it starts moving upward and evenmaUy crosses the real axis for Sx = 1.65. For... 

Fig. 4.33 The scan impedance atf = 9 GHz when the array in Fig. 4.32c is scanned in the E piane. Broadside starts at s> = 0 and grazing is at Sx= 1.0. Higher vaiues of s correspond to scan in the imaginary space where the scan impedance is purely Imaginary. It gets to the "end of imaginary space for Sx = 2.08 from where it goes right back down the way it got in.

Fig. 4.34 Same scan impedances as shown in Fig. 4.33 but atf = 8 and 10 GHz, respectively. Note how the higher frequencies move up toward the inductive region of the complex plane and vice versa for the lower frequencies. At some frequencies between 8 and 10 GHz we have the possibility to pass close to the origin once and even twice.

So far we have considered surface waves only on finite periodic structures without a groundplane. When a groundplane is added to an array of dipoles, it is usually driven actively. This case is in practice somewhat different from the passive case considered above by the fact that aU elements are connected to generators or amplifiers with impedances comparable to the scan impedances. As explained in Chapter 5, this leads to a highly desirable attenuation of any potential surface waves. 

The determination of the locus for the scan impedance for a single slab in Fig. 6.10 and a double slab in Fig. 6.11 is only qualitative. However, the reduction of VSWR and to some extent an increase in bandwidth in the double-slab case is quite obvious. Inspection of the figures of the two cases shows that the primary reason is that the double-slab case produces a smaller variation of the real part of the scan impedance while the imaginary part must be balanced very carefully with the groundplane reactance Z2+ and the antenna reactance In other words, a parametric study of actual calculated curves should be undertaken. A typical example will be given in the next section. 

CALCULATED SCAN IMPEDANCE FOR ARRAY WITH GROUNDPLANE AND TWO DIELECTRIC SLABS... 

Finally, we show in Fig. 6.13 the scan impedance for scanning at 30°, 45°, and 60°, respectively, in the H plane, while Fig. 6.14 shows the f -plane scan at the same scan angles. Compared to the single-slab cases shown in Figs. 6.8 and 6.9, we observe that the double-slab case continues to be superior to the single-slab... 

Fig. 6.13 The scan impedance for the same case as shown in Fig. 6.12 but for scan angles equal to 30°, 45°, and 60° in the H plane.

In addition, it is well known that a dielectric slab reduces the variation of the scan impedance with scan angle (see reference 117). 

It is by now well known that the variation with angle of incidence of the scan impedance of phased arrays as well as the bandwidth of hybrid radomes can be reduced by using dielectric slabs placed between free space and the device in question. To be sure, the dielectric constant should in general be less than 2 (for a single slab) and the thickness should be somewhat thicker thau A./4 in the dielectric. An example of applying this technique is shown in Fig. C.15. Compared to the uncompensated case in Fig. C.13, we observe some improvement... 

When working with phased arrays, it is of particular interest to compare the scan impedance with the embedded impedance, in particular whether one can be derived from the other. 

The first of these, namely the scan impedance, is the impedance observed at the element terminals when the proper voltages are applied to all the elements. These voltages may all have the same amplitude corresponding to uniform aperture illumination or they may be tapered across the aperture. When we vary the phases of these terminal voltages in a linear fashion across the aperture, the beam will scan in different directions and the scan impedance will in general vary considerably with scan angle. 

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