impedance - Converting S parameters to Z parameters - divergence

I have cause to convert some Z parameters to S parameters and vice versa. Here is the conversion for $Z_{11}$;

$$Z_{11} = {((1 + S_{11}) (1 - S_{22}) + S_{12} S_{21}) \over \Delta_S} Z_0$$

where

$$\Delta_S = (1 - S_{11}) (1 - S_{22}) - S_{12} S_{21}.$$

I thought I'd start nice and simple with an ideal short transmission line of impedance $Z_0$. Now $S_{11}=0$, $S_{21}=1$, $S_{12}=1$ and $S_{22}=0$. Therefore $\Delta_S=0$, so $Z_{11}$ diverges (as do the other Z parameters) and something has gone wrong somewhere.

Where have I messed up?

Answer

You didn't mess up.

$Z_{11}$ is the input impedance when the other port is terminated with an open circuit.

Since your device is just a bit of wire, it has infinite input impedance when the other end is not connected to anything, and thus infinite $Z_{11}$.

This is an example of why we need different two port representations (S-parameters, Y-parameters, Z-parameters, H-parameters). There's certain devices that can't be represented in any particular representation.