3 Disc. Sim. of Cont. Systems

The following table summarizes three different approximate mappings between the s and z planes.

Method Approximation Inverse Mapping
Forward Rectangular $s\approx\frac{z-1}{T}$ $z\approx 1+Ts$
Backward Rectangular $s\approx\frac{z-1}{Tz}$ $z\approx\frac{1}{1-Ts}$
Trapezoidal Rule $s\approx\frac{2}{T}\frac{z-1}{z+1}$ $z\approx\frac{1+Ts/2}{1-Ts/2}$

 

The following figure shows how each approximation maps the left-half s-plane into the z-plane.

s and z plane mapping

Figure 1: Contours in s and z planes. Shaded region correspond to mapping from the s-plane to the z-plane. Click-and-drag the mouse in the s-plane to see mappings in the z-plane

In Figure 1, select an integration type, then draw on the s-plane graph. The points, corresponding to the selecting mappping, will be shown on the z-plane. Notice the mapping at the stability boundary by drawing (click-and-drag) points along the imaginary axis on the s-plane. Notice there is no scaling on the s-plane so be careful about drawing conclusions.

  1. How does the stability boundary on the s-plane (the imaginary axis) map to the z-plane for backward and trapezoidal integration? Pay attention to the linearity of the mapping.
  2. The backward integration never maps to the negative real side of the z-plane. What does that say about the response of the corresponding z transforms?
  3. Select the trapezoidal integration and draw points along the real axis. As some location the point maps to (0,0) on the z-plane, or a pole at $z=0$. Speculate as to what the means.