S11 of Ideal Transmission line on a Smith Chart

[Xyius]

Homework Statement

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?

Homework Equations

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

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?

 

[Daz]

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