I'm trying to figure out how radio frequency "matching stubs" work. In order to fully understand the problem, I need to know how the "curve of equal SWR" looks like.
I did a few plots, and it looks like it's a circle, or something close.
Whenever I approach the matter analytically, the equations explode and become very ugly.
I hope there's an easy way to prove that the following equations describes a circle, or ellipse on the complex plane:
$VSWR = \frac{1+|\Gamma|}{1-|\Gamma|} $
where VSWR is a constant, and $\Gamma$ (capital gamma) is the reflection coefficient:
$\Gamma = \frac{Z_{load}-Z_{line}}{Z_{load}+Z_{line}}$
$Z_{line}$ is a constant, normally a real number, actually equals to 50 Ohms in my specific case. (However, it would be interesting to see what happens if this can be a complex value, but this is really unimportant right now).
$Z_{load}$ is the variable, I suspect that these can be found on a circle or ellipse on the complex plane.
So can you girls/guys see an easy way to prove that $Z_{load}$ values are on a circle or ellipse? I do not seek absolute precise mathematical proof (it would be nice though), I just want to be "reasonably sure" that this is ture.
Thank you very much,
Tamás.