S11 of Ideal Transmission line on a Smith Chart

1. Sep 29, 2015

Xyius

1. The problem statement, all variables and given/known data
An ideal transmission line is terminated with a load impedance of 90 ohms and has a characteristic impedance of 50 ohms. Why does it's S11 response trace out a circle on the smith chart?

2. Relevant equations
$$Z_{\text{in}}=Z_0\frac{Z_0+j Z_L \tan(\beta l)}{Z_L+j Z_0 \tan(\beta l)}$$
$$S_{11}=\Gamma=\frac{Z_{\text{in}}-Z_0}{Z_{\text{in}}+Z_0}$$

3. The attempt at a solution
So my thought was that perhaps I can plug the expression for $Z_{\text{in}}$ into the expression for $S_{11}$ and simplify and get an expression for a circle. After many tries, the closest thing I can come up with is the following.

$$S_{11}=\frac{e^{-j\beta l}}{(Z_0+Z_L)}\left[ (Z_L-Z_0)\cos(\beta l)- j (Z_L+Z_0) \sin(\beta l) \right]$$

This kind of looks like the equation for an ellipse inside the brackets, but the term out front kills it.

Am I doing this all wrong???

2. Oct 3, 2015

Daz

If you set l, the length of the transmission line to zero, then ZIN should equal ZL. But it doesn't in your first expression. Look again at your expression for ZIN.