Mapping of S plane to Z plane

The S plane is used for analysis of continuous signal analysis and design where as the Z plane is used for discrete signal analysis.

S = σ+jω, Real and the imaginary part ............(1)

Z = e^st.....................(2)

putting the value of equation in z we get

 

Hence 

 (Periodicity)

For stable system, σ is negative in the S domain.

So that magnitude of Z is less than 1.

|z| = e^Tσ < 1

S Plane

Z Plane

Lets see an example for plotting poles in both these graphs.

S plane                                                            Z plane

To understand how the Behavior of this stability pattern, see this video. Here.

To understand the Mathematical relation between both S and Z planes see this video. Here.

Thanks & see you in the next one...