Complex Mapping Theorem

 I. Preliminaries:

  a) Complex Mapping: Given a rational polynomial function,


  G(s) = num(s) / den(s)

 where num(s) is the numerator polynomial in s,
and den(s) is the denominator polynomial in s.
Then if s is a complex number, G(s) is also a complex number, and the value of s is said to be mapped from the s-plane to the G(s)-plane.
Ex. G(s) = 1/(1.5s+1). Let s = 1+2i, G( 1+2i ) = 1/( 1.5*(1+2i) + 1 ) = 0.1639 - 0.1967i
 
b) A simple closed path G is one which starts and ends at the same point without crossing itself.


II. Complex mapping theorem:

Given: 1) A rational polynomial function, G(s), and
          2) A simple closed path G in the s-plane which does not pass through any poles or zeros of G(s).

Then: As s traverses the path G in the clockwise direction, the map G(s) will encircle the origin N times in the clockwise manner, such that

  N = Z - P
 where Z is the number of zeros of G(s) inside G, while P is the number of poles of G(s) inside G.
Example: G(s) = ( s2+3s+8 )/( s2+3s+2 )
 zeros: -1.5 + 2.4i, -1.5 - 2.4i      poles: -2, -1
  G is the path A-B-C-D-A, as shown in figure below:

Figure 1.


Since Z = 0 and P = 2, the complex mapping theorem predicts N = 0-2 clockwise encirclements, or 2 counterclockwise encirclements of the origin.

To show this is true, we use the following evaluations,

 
S
G(s)
Symbol
A
1.0 + 2.0i
1.1154 - 0.5769i
*
 
1.0 + 1.0i
1.6000 - 0.6000i
*
 
1.0
2.0000
*
 
1.0 - 1.0i
1.6000 + 0.6000i
*
B
1.0 - 2.0i
1.1154 + 0.5759i
o
 
0.2 - 2.0i
0.8303 + 0.8484i
o
 
-0.6 - 2.0i
0.1675 + 0.8712i
o
 
-1.4 - 2.0i
-0.4026 + 0.1323i
o
 
-2.2 2.0i
-0.0265 0.7644i
o
C
-3.0 2.0i
0.7000 0.9000i
x
 
-3.0 1.5i
0.9262 1.3292i
x
 
-3.0 1.0i
1.6000 1.8000i
x
 
-3.0 0.5i
2.9765 1.6941i
x
 
-3.0
4.0000
x
 
-3.0 + 0.5i
2.9765 + 1.6941i
x
 
-3.0 + 1.0i
1.6000 + 1.8000i
x
 
-3.0 + 1.5i
0.9262 + 1.3292i
x
D
-3.0 + 2.0i
0.7000 + 0.9000i
+
 
-2.2 + 2.0i
-0.0265 + 0.7644i
+
 
-1.4 + 2.0i
-0.4026 0.1323i
+
 
-0.6 + 2.0i
0.1675 0.8712i
+
 
0.2 + 2.0i
0.8303 0.8484i
+
A
1.0 + 2.0i
1.1154 0.5769i
*

The plot in Figure 2 verifies that there are indeed 2 counterclockwise encirclements of the origin.
 
 

FIGURE 2.





 

This page is maintained by Tomas B. Co (tbco@mtu.edu). Last revised 2/8/1999.
     Tomas B. Co
     Associate Professor
     Department of Chemical Engineering
     Michigan Technological University
     1400 Townsend Avenue
     Houghton, MI 49931-1295

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