Termination impedance

Deviating from conventional installations (in many cases characterized by several field devices connected to a common terminal box/junction box), the fieldbus cable with its intrinsically safe circuit passes from transmitter to transmitter (Fig. 6.213). Each transmitter is fitted with a branch box (T box) to connect it to the fieldbus (Fig. 6.214). At the physical end of the bus segment, a terminating impedance shall be fitted. 

In passing, we note that this device can be made into a power divider/combiner merely by terminating the A port and having the rf input at B. Alternatively, an out-of-phase power divider/combiner can be made by putting the terminating impedance on B and the input at A. In fact, power dividers are made in exactly this way with the terminating impedance internal to the package. 

In the first case we will observe a significant increase in the bistatic scattering. In the second case we will observe a variation of the terminal impedance as we move from column to column. Let us look upon these two phenomena separately. 

But how do we explain the much stronger reduction of the ripples associated with the surface waves Well, we shall later in Chapter 4 investigate surface waves in much more detail. It will there be shown that the terminal impedance associated with the surface waves is quite low, say of the order of Zsarj 10 ohms for each of the two surface waves. Thus, by the same reasoning as for the Floquet currents above, we find for each surface wave a reduction equal to 10/(10 + 100) = 0.091. This is of course an average value but explains the strong ripple reduction observed in Fig. 1.5a. 

In Chapter 1 we introduced the fundamental concepts concerning a new type of surface wave that can be excited only on finite periodic structures. It was pointed out that radiation could occur from such surface waves and therefore could lead to an increase in the RCS level in the backward direction. Similarly, if the structure was active—as, for example, for a phased array—this type of surface wave could lead to a very significant variation of the terminal impedance form element to element. This could make precise matching difficult, if not impossible. 

While such an approach is feasible, it should be applied with great care. A very simple example will illustrate what can easily go wrong and undetected by operators whose intellectual capacity is limited to comparing numbers. In Fig. 5.37a we show the typical backscattered fields in vector form from each triad similar to the case in Fig. 5.8. Recall that the fields scattered from the two edge triads are quite different from the rest if all the triads are loaded with identical load resistors Rl- This was simply because the terminal impedances of the edge triads were in a different element environment, resulting in terminal impedances different from the rest. 

The second problem is actually more complex. When considering an infinite array, the terminal impedance will be the same from element to element in accordance with Floquet s Theorem. However, when the array is finite, it is well known that the terminal impedance will differ from element to element in an oscillating way around the infinite array value (sometimes denoted as jitter). We postulated that this phenomenon was related to the presence of surface waves of the same type as encountered in Chapter 4. However, there is a significant difference in amplitude of these surface waves in the passive and active cases. This is due to the fact that the elements in the former case in general are loaded with pure reactances (if any), while the elements in the latter case are (or should be) connected to individual amplifiers or generators containing substantial resistive components (as encountered when conjugate matched). 

These resistive components cause significant attenuation of potential surface waves along the structure. In fact, they will in general be so weak that the surface wave radiation from active arrays can be ignored in contrast to the FSS case discussed in Chapter 4. However, they may be strong enough to produce jitter of the terminal impedance. 

Fig. 6.2 Typical terminal impedance Za = 2Rao + S a be negative direction. The interelement spacing is varied from Dx/X = 0.75 to 0.25. The groundplane impedance Zr+ is purely imaginary that is, it is located on the rim of the Smith charts as shown.

Fig. 6.5 Top The equivalent circuit for an array of wire dipoles backed by a groundplane. Bottom The groundplane impedance at the rim of the Smith chart is being connected In parallel with 2Rao to the left and denoted 2Rao Zi+. Finally, adding the antenna reactance JX in series is seen to produce a more compact terminal impedance 2Rao Zi+ + jX than without a groundplane.

Fig. 6.6 Top The equivedent circuit for an array of wire dipoles with a dielectric slab in front and a groundplane in the back. Bottom From the array we look left through the dielectric slab and see Zr. Then we connect the groundplane impedance Zi+ in parallel and obtain Zi- Z,+. Finally, we connect the antenna reactance jX in series and obtain the compensated terminal impedance curve Zr Zi+ +P(a-...

The performance of a typical high frequency fn is obtained completely analogous to the fi case. If jX is symmetric around fo (which is idealized), we will simply find that the terminal impedance is given by the point fn located symmetrically to fi- The typical terminal impedance curve as a function of frequency is denoted by Zi Zi+ - - jX in Fig. 6.6. 

Fig. 6.7 Calculated terminal impedance at broadside obtained from the PMM code based on the dimensions shown in the insert. The values include a small matching section comprised of a transmission line of length 0.13 cm and characteristic impedance 200 ohms.

Fig. 6.10 Single dielectric slab. The input impiedance as seen through the dielectric slab at the array in the negative direction is denoted Zi-. As shown in the schematic, we then connect the groundplane impedance Z + in parallel and obtain Zi- Z). Finally, we add the antenna reactance JXa in series and obtain the comp isated terminal impedance cunre Zi. Zf++/Xy,.

Fig. 6.11 Double dielectric slab. We first obtain the input impedance Zi- of slab 1 looking left. Next, we find the input impedance Z2- of the second slab when terminated in Zi— From the schematic we then see that the ground plane impedance Zj+ should be added in parallel yielding Z2- IIZ2+, Finally, we add the antenna reactance jX in series and obtain the compensated terminal impedance Z2- Z2+ + jX. ...

Today we know that it is actually easier to calculate the terminal impedance of an infinite array than it is to calculate it for a single element. And the mutual coupling should not be viewed as an evil to be avoided. On the contrary, when handled properly, it can be a true blessing to be enjoyed. And you certainly cannot outlaw it. 

We have so far considered RCS in-band reduction only. The success of this approach depends strongly upon how well the antenna is matched to the load (i.e., the terminal impedances of receivers or transmitters). Since a good match... 

B.3 Design a broadband matching network for the terminal impedance shown in Fig. 5.36, bottom. This can actually be done simpler than the example treated in Fig. B.3. You may land the matched impedance anywhere on the real axis between 150 and 260 ohms. 

The embedded impedance, on the other hand, is the terminal impedance observed at just one element usually located somewhere in the middle of the array and with all the other elements terminated in loads (usually resistive). Thus, a terminal voltage is only applied to a single element, while all the other elements are excited parasitically. 

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 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. 

We finally show in Fig. D.ll the terminal impedance as seen from the center element with two parasitic sections at the top and two below like Fig. D.7b but also flanked on each side with parasitically excited columns comprised of five segments. Dimensions are given in the insert, and the ESP program was used with a 10-mil dielectric sheet. As usual, the impedance curves include the small transmission line section. Note the poor performance at the lower frequencies compared to the scan impedance in Fig. D.3. 

Figure 3.1.5. Schematic diagram of a working (modified) Berberian-Cole bridge shown as a four-terminal impedance-measuring system.

Because of the potential distribution along the crack, this equivalent circuit should more properly be represented as a nonuniform, finite transmission Une for the oxide wall impedance, with the crack tip as a terminating impedance. Such a case is shown in Figure 4.4.46. 

This bulky equation reduces to simple and familiar equations for situations where the terminating impedance is infinite (when pores are terminated by a wall) or when it is zero, as in the case of conduction through a membrane. Note that these equations for conduction through a porous layer are mathematically identical to those for diffusion, given in Chapter 2.1.3.1... 

Eq. (12), is one where electrodes and diffusion in the electrolyte between electrodes are represented by means of an ion-producing electrode in series with a transmission line, terminated with an electrode that consumes diffusing ions. In this case, the terminating impedance Zt will be the impedance of this electrode. An alternative treatment of diffnsion with imperfect boundary conditions is given in Franceschetti et al. [1991]. 

The only condition under which no reflection takes place and the pulse is absorbed completely is when the terminating impedance is equal to the characteristic impedance of the cable (i.e. = Zg). In effect, the pulse... 

---