s to z parameter matrix

In such a case why worry about introducing s parameter matrix and so on ..smith chart...etc?!

one can directly apply the formula for getting s parameters once he knows Z and U matrix....

It depends on your purpose whether you need S parameter or Z-parameter. If you want to look at the reflection why do you want to calculate Z first? Again, if you want to see resonance, it is better to see the Z-parameter (however, smith chart gives you the resonance).

One more point, many characteristics of HF are clear from smith chart, which cannot be directly seen from Z parameters.

"Microwave Transistor Amplifiers, Analysis and Design" , Guillermo Gonzales ; section 1.9 relates the various scattering parameters. It has the relationships you seem to be seeking.

(I have only paper copies)

Hello
S=(Z-R)*(Z+R)^-1
Where Z-z-matrix, R-diag real matrix of wave impedances.