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┏━┓
┃  ┃
┗━┛
Z
a) F(s) -s - zi
           
           
┏━━┳━┓
┃jco ┃  ┃
┃-/  ┃  ┃
┗━━┻━┛
0
Z
  o
b) F(s) = 1/(s - pi)
             
  ~   }----  
┏━┳━━━┓
┃o ┃~o    ┃
┣━╋━━━┫
┃  ┃      ┃
┗━┻━━━┛
           
┏━━━━┓
┃        ┃
┣━━━━┫
┃      - ┃
┗━━━━┛
┏━━━━┳━━━━━━━━┓
┃  Im d  ┃ L              ┃
┃(       ┃/-       '  R*  ┃
┃(       ┃                ┃
┃~-      ┃                ┃
┣━━━━╋━━━━━━━━┫
┃        ┃                ┃
┗━━━━┻━━━━━━━━┛
c) F(s) : s - zi
Fig.5.17   nlustration ofcontour mapprng.
 corresponding points in the F-plane and obtain the contour B. Then the contour A
in the s-plane is said to be mapped to the contour B in the F -plane. We assume
 that the mapping is one to one, i.e., for every point in the s-plane, there is one and
 only one corresponding point in the F -plane and vice versa.
       To understand the concept ofmapping further,let F(s)  - s - Zi  and let the point
 s  - zi, which is the zero of F (s), lie outside the contour A as shown in Fig. 5.17a.
Instead of using the coordinates of point P,let us use the vector approach. Every
point P on the contour A is associated with a vector V. Let V' be the image
vector in the F-plane. For this case, IV'I = IVI and Z V' - / V. Now as we move
clockwise along the contour A, the magnitude and phase of the vector V varjr.
The phase oscillates betwcen.the two limiting values ~i and @z. In this case, a
clockwise movement along the contour A corresponds to a clockwise movement
along the image contour B in the F-plane.
     Now let F(s) = l/(s - pi) and let the pole s = Pi lie outside the contour A as
shown in Fig. 5.17b. For this case,  IV'I  =  1/I VI  and  L V'  =  -Z V. As a result, the
contour A in the first quadrant maps to contour B in the fourth quadrant. Observe
LINEAR SYSTEMS, THEORY, AND DESIGN: A BRIEF REVIEW       475
                 
  j(D d   /---   
┏━━━┳━━━┓
┃(     ┃      ┃
┃0     ┃~/y - ┃
┣━━━╋━━━┫
┃      ┃      ┃
┗━━━┻━━━┛
               
┏━┳━━━━┓
┃  ┃0r ~-   ┃
┃\ ┃~-,) ', ┃
┣━╋━━━━┫
┃  ┃        ┃
┗━┻━━━━┛
d) F(s) - l/(s - pi)
                       
        J(D J          
┏━━━━━━┳━━━┓
┃\           ┃~-y ' ┃
┣━━━━━━╋━━━┫
┃            ┃      ┃
┗━━━━━━┻━━━┛
                           
  Im .                     
┏━━━┳━━━━━━━━┓
┃      ┃                ┃
　
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