6.14 Single stub matching

Often, there are reasons why using a discrete inductor or capacitor for matching is not such a good idea. Atthe high frequencies where matching is important, losses in both L or C mean that you don't get a good match, and most of thetime (except for some air-dielectric adjustable capacitors) it is hard to get just the value you want.

There is another approach though. A shorted or open transmission line, when viewed at its input looks like a purereactance or pure susceptance. With a short as a load, the reflection coefficient has unity magnitude $\left|\Gamma \right|=1.0$ and so we move around the very outside of the Smith Chart as the length of the line increases or decreases, and $\frac{Z_{\mathrm{in}}}{Z_0}$ is purely imaginary. When we did the bilinear transformation from the $\frac{Z(s)}{Z_0}$ plane to the $r(s)$ plane, the imaginary axis transformed into the circle of diameter 2, which ended up being the outside circle whichdefined the Smith Chart.

Thus, instead of a discrete component, we can use a section of shorted (or open) transmission line instead . These matching lines are called matching stubs . One of the major advantages here is that with a line which has an adjustable short on the end ofit, we can get any reactance we need, simply by adjusting the length of the stub. How this all works will become obviousafter we take a look at an example.

Let's do one. In we can see that, $\frac{Z_L}{Z_0}=0.2+0.5i$ , so we mark a point "A" on the Smith Chart. Since we will want to put the tuning or matching stub in shunt across theline, the first thing we will do is convert $\frac{Z_L}{Z_0}$ into a normalized admittance $\frac{Y_L}{Y_0}$ by going ${180}^{°}$ around the Smith Chart to point "B", where $\eqsim (\frac{Y_L}{Y_0}, 0.7+-1.7i)$ . Now we rotate around on the constant radius, $r(s)$ circle until we hit the matching circle at point "C". This is shown in . At "C", $\frac{Y_S}{Y_0}=1.0+2.0i$ . Using a "real" Smith Chart, I get that the distance of rotation is about $0.36\lambda$ . Remember, all the way around is $\frac{\lambda }{2}$ , so you can very often "eyeball" about how far you have to go, and doing so is a good check on making a stupid matherror. If the distance doesn't look right on the Smith Chart, you probably made a mistake!