Transformation circles

In the following, several cases will demonstrate how to construct the transformation circles given various parameters. 

Fig. A.2 Case II Given the load impedance Z and the characteristic impedance Zj as points in the complex plane (troth real). The low impedance Zt is obtained by drawing the tangent from Zl to the circle z = Zt ei. Next, project the touching point A upon the real axis to obtain point B, which determines Zt. The points ZJ and Zt, = Zl complet y determine the transformation circle going through Zl. Note Only valid for the rectangular coordinate system.

Fig. A.3 Case III Given Zl (arbitrary) and the characteristic impedance Zi on the real axis. Draw a tangent from Zl to the circte z= jZtjei and project it onto the line OB r producing point B. The transformation circle for Zm will go through B and B-t with center on the real axis (for Z real). This approach is only valid in the rectangular coordinate tem.

A.3 WHERE IS Zi LOCATED ON THE TRANSFORMATION CIRCLE DETERMINATION OF THE POSITION CIRCLES... 

So far we have concentrated merely on determining the transformation circle for Zj . In this section we discuss how to determine where on that circle Z is located. Since this subject is thoroughly described in reference 123, only a brief overview of the approach is given. 

The admittance case is shown in row 2. It follows directly from the impedance case in row 1 by the simple inversion Y = 1/Z, that is, circles will be transformed into circles. More specifically, the characteristic admittance is Ti = 1/Zi while the transformation circle is determined by the extreme points... 

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. 

The beauty of this situation becomes clear when we rotate curve 2 along the transmission line of length 24.6 cm. Since the rotations along the transformation circles are proportional to the frequency, the higher frequencies will travel further. However, since they are behind the lower frequencies (by virtue of reversing Zi into curve 2), it becomes clear that curve 2 might be clustered together as they arrive at point 3. We observe this to be the case indeed however, curve 3... 

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... 

The conclusion of the discussion above is that curve 2 should be manipulated to produce transforming circles as close to each other as possible. That is not likely to land curve 3 at the right impedance level. This can subsequently be corrected by use of one or more transformations as explained above. Alternatively, other tools from Fig. B.l can be used as discussed next. 

Furthermore, the curve depicting Zl should be shaped or curved the proper way. More precisely, after it is reversed (for example) as illustrated above, it should ideally follow a transformation circle as close as possible. This turns out to be one of the more intriguing tasks in broadband matching. Typically, the... 

A.3 Where is Z Located on the Transformation Circle Determination of the Position Circles / 286... 

---