Smith chart

Fig. 13. a) A schematic of the LC-tank circuit used to transform the high impedance of the RF-SET down to the characteristic impedance of Z = 50 Q. h) Using the simulation software ADS, Sn is plotted for the case when Rrf-set = 100 kQ (hrown) and when it is 130 kQ (green) along with the corresponding Smith chart, (c). The simulated component values of the tank circuit were Cpamsitk =0.18 pF, Ltank = 710 nH, and the resistance of the inductor at its resonant frequency Rk-rank = 10... 

More specifically, consider Fig. 2.6, top. We show here a Smith chart normalized to Ra. Recall farther that one of the nnique features of the Smith chart is that the reflection coefficient F as given by (2.2) is a phasor measured from the center Ra of the Smith chart to the modified load impedance Rp + jiXi + Xa) s shown in the Smith chart. The magnitude of the field associated with the antenna mode component is proportional to r [see (2.8)], while the magnitnde of the field associated with the residual mode component is proportional to C [see (2.10)]. Thus, according to (2.9) the field associated with atot is proportional to the magnitude of the phasor sum (T + C) as also shown in the same Smith chart. 

In Fig. 2.8 we show a dipole array seen edge-on under three different load conditions. The element lengths are A./2. In the case to the left the terminals are open-circuited. Thus, looking into the antenna terminals we observe a reflection coefficient F = -t-1 as also shown in the Smith chart underneath the array. 

Finally we consider the case to the right in Fig. 2.8. We have here loaded the array at its terminals with short circuits thus F = — 1 as indicated in the Smith chart underneath the array. The total backscattered field is proportional to F - -C —1 — 1 = —2 [see (2.9)]. However, when viewed as an FSS of elements with length 21 A./2, we also know that such a surface reflects as a groundplane. Thus the reflection coefficient for an incident wave is Ff s = — 1. In other words, F - -C = —2 produces a reflection coefficient equal to Ffss = —1, and consequently the matched case in the middle with C = — 1 will produce a reflection coefficient equal to Ff s = — 1 /2. 

Our next step is inspired by the Smith chart. (Some readers are happy to see this ingenious device back in in our curriculum. It was never out as far as I am concerned. See Chapter 6 and Appendices A and B and you will understand my devotion.). 

Advantage of using Smith chart It tells us HOW to adjust Zl to obtain 0. 

Fig. 5.7 A plane wave incident upon an infinite array of active elements in front of an infinite groundpiane. By plotting the reflected field in a complex plane (/n this case a Smith chart) we can adjust the load impedances Zrofthe active dipoles such that the reflected field disappears in the backscatter direction.

However, while the adjustment of Zl can be done precisely in the Smith chart, this is not quite the case for F shown in Fig. 5.6. One reason being that the field reflected from the infinite array is a simple plane wave (we assume the evanescent waves have died out), while the fields from the triads are a combination of Hankel functions. Should we call this new chart a Hankel chart Somebody could work it out and cover himself with fame and glory. See also comments in Section 5.8. 

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. 6.15 Typical antenna impedance = 2Rao +JXa in the negative direction for a long wire antenna like a flat spiral without the effect of a groundplane. At lower frequencies the groundplane impedance Zi+ is inductive as shown on the rim of the Smith chart. The total antenna may improve if jXa is capacitive but get worse If Inductive. Thus, no broadband compensation will take place.

We may further assume that the FSS is transparent at /i = 1 GHz (this is a gross approximation, but since this scheme is a scam anyway, so what ). Under that assumption we obtain Z + = oo at /i = 1 GHz as indicated in the Smith chart in Fig. 6.16. (A more rigorous calculation will show that the distance between the array and the groundplane should be somewhat less than A.i/4 when we take into account the capacitive effect of the FSS at /i = 1 GHz.)... 

Similarly, Zi+ = oo at /2 = 2 GHz as also indicated in the same Smith chart. So far we have obtained our objective Zi+ is infinite at 1 and 2 GHz. The question now is. What will happen at other fi equencies First of all we note that aU components to the right of the array are lossless. Thus, Zi+ must be purely imaginary—that is, located somewhere on the rim of the Smith chart. It is now a weU-established fact that all impedances at the rim of a Smith chart will move clockwise with increasing frequency (it cannot just stop ). Thus, when going from 1 to 2 GHz we simply must follow the rim of the Smith chart that is, at some frequency between 1 and 2GHz, Zi+ must pass through zero in the Smith chart, which of course is a clear disaster. (For those who need a more rigorous proof, consult Foster s Reaction Theorem [101].)... 

Another design had placed a circuit analog sheet less than O.OIA in front of the groundplane. Even an inexperienced designer should know that a groundplane that close to a CA sheet basically shorts it out, leaving it virtually ineffective (try it out in a Smith chart and you will soon understand). 

The Smith chart is weU-suited for handling transmission line problems when normalized to the characteristic impedance of the transmission line in question. However, when working with several transmission line sections with different characteristics impedances, for example, we must renormalize the Smith chart each time a change is made to a new characteristic impedance. Since all Smith charts look alike after normalization, we can easily lose track of just exactly where we are when dealing with a complex matching problem. 

Fig. A. 1 Case I Given the ioad impedance Z and characteristic impedance Zi, numericaiiy caicuiate r, Z/ and Zh from the formuias in the figure. This compieteiy determines the transformation circie going through Zl. Note This approach is vaiid for Smith charts as well, as long as we read all the impedances in the Smith chart.

Fig. A.4 Case IV Given two arbitrary points 82 and 83, the transfonvation circle for Zi can be drawn immediately with center on the real axis assuming that the characteristic impedance Zi is real. Next draw a tangent from the origin O touching the transformation drcle at point T. The length ofOT = Zi. This approach is only valid in the rectangular coordinate system unless actual numbers are read in the Smith chart.

Figure B.l is organized as follows. In row 1 we show the three cases depicted at the top in the impedance plane Z in rectangular coordinates. Next, in row 2 we show the same three cases in the admittance plane Y in rectangular coordinates. Similarly, we show in row 3 the impedance plane Z like row 1, but this time depicted in a Smith chart normalized to an arbitrary impedance Zq. Finally, row 4 shows the admittance plane Y, as in row 2, but depicted in a Smith chart normalized to an arbitrary admittance Yo = l/Zo-...

In row 3 we show the impedance plot from row 1 but now plotted in a Smith chart normalized to Zq. Again, we notice that the transformation from row 1 to row 3 is bilinear [123] that is, general circles will be transformed into general circles. In particular, the imaginary axis in row 1 will be transferred into the rim of the Smith chart in row 3 while the real axis will be transformed into itself. Furthermore, the two lines X = 7X1 and 7X2 will have their infinity point transformed into 00 in the Smith chart and the two points 7X1 and 7X2 are located on the rim of the Smith chart. We further note in row 1 that the angle between the two horizontal lines and the real axis is zero that is, the center for... 

One might at this point wonder Why consider the general Smith chart case where Zi Zq Let us emphatically state that actual detailed calculations in a Smith chart should not in general be performed unless Zi = Zo. However, as will be illustrated later, a typical situation occurs when Zi is not a single point in the Smith chart but actually a given curve obtained by measurements (or otherwise) and plotted in a Smith chart normalized to Zq. It is in that case quite convenient to quickly draw the transformation circle for the extreme points of Zl for various test values of Z. See Appendix A. As discussed later, that will tell us what the smallest possible VSWR can be for a given curve for Zl. See Section B.6. 

Given The load impedance Zl as a function of frequency is shown in a Smith chart normalized to both 200 ohms and 100 ohms as shown in Fig. B.2. We are required to design a single series stub tuner as shown in the insert that transforms Zl at the center frequency / = 250MHz into 50 ohms. Based on that design at the center frequency we are next required to find the input impedance Zi at the other frequencies. 

The other frequencies are determined by noting that their rotation in the Smith chart is proportional to the frequency—that is, 0.129 [(//250A)]. Furthermore, we shall assume that the series reactance jX varies with frequency like an ideal lumped load that is, it is also proportional to the frequency (stubs of transmission lines do not behave quite that simply). The final result is seen in the Smith chart normalized to Zq = 100 in Fig. B.2. The VSWR circle corresponding to 2 with respect to 50 ohms is also shown (see Problem B.2). Needless to say, broadbanded it ain t ... 

The typical problem with that suggestion might be that although the new transformation circles based on 160 ohms may appear in the figure to be as much apart from each other as the ones associated with 200 ohms, they might in effect lead to a curve 3 that is relatively wider than before because of the nature of the Smith chart (see also Problem B.2). But a more severe problem might be that the... 

An undercompensated curve 3 can also be compressed by an insertion of a series circuit or equivalent as shown in Fig. B.7 to the left. At the lower frequencies the series circuit is capacitive, which will move the lower frequencies of curve 3 toward the capacitive part of the Smith chart along the constant resistance circles (see Fig. B.l, row 3, column 1). 

We may conclude from the discussion above that quite a considerable time and effort is often spent to manipulate the load impedance to be just right, or as it is sometimes expressed in the matching jargon, Make sure it is bom right. No question about it, experience plays a significant role in that development and there is no better way than to fight it out partly by hand in a Smith chart. Once a concept is developed, the fine-tuning can be done on the computer. 

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