Module 2 : Modeling Discrete Time Systems by Pulse Transfer Function

Lecture 2 : Relationship between s-plane and z-plane

Since

$\displaystyle e^{(s+jmw_{s})T}$
 $\displaystyle =$ $\displaystyle e^{Ts}e^{j2\pi m}$  
  $\displaystyle =$ $\displaystyle e^{Ts}$  
  $\displaystyle =$ $\displaystyle z$  

where m is an integer, all the complementary strips will also map into the unit circle.

1.1 Mapping guidelines

  1. All the points in the left half s-plane correspond to points inside the unit circle in z-plane.
  2. All the points in the right half of the s-plane correspond to points outside the unit circle.
  3. Points on the jw axis in the s-plane correspond to points on the unit circle $ \vert z\vert=1$ in the z-plane.
    $\displaystyle s$
    		 $\displaystyle =$ $\displaystyle jw$  
    $\displaystyle z$
    		 $\displaystyle =$ $\displaystyle e^{Ts}$  
      $\displaystyle =$ $\displaystyle e^{jwT} \Rightarrow$    magnitude =1  

 

1.2 Constant damping loci, constant frequency loci and contant damping ration loci

Constant damping loci: The real part σ of a pole, $ s=\sigma+jw$, of a transfer function in s-domain, determines the damping factor which represents the rate of rise or decay of time response of the system.

. Large σ represents small time constant and thus a faster decay or rise and vice versa.

. The loci in the left half s-plane (vertical line parallel to jw axis as in Figure 2(a)) denote positive damping since the system is stable

. The loci in the right half s-plane denote negative damping.

. Constant damping loci in the z-plane are concentric circles with the center at z = 0 , as shown in Figure 2(b).

. Negative damping loci map to circles with radii >1 and positive damping loci map to circles with radii <1.