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 splane 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 splane 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
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) = ( s^{2}+3s+8 )/( s^{2}+3s+2 )
zeros: 1.5 + 2.4i, 1.5  2.4i poles: 2, 1
G is the path ABCDA, as shown in figure below:
Since Z = 0 and P = 2, the complex mapping theorem predicts N = 02 clockwise encirclements, or 2 counterclockwise encirclements of the origin.To show this is true, we use the following evaluations,




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


0.2  2.0i

0.8303 + 0.8484i



0.6  2.0i

0.1675 + 0.8712i



1.4  2.0i

0.4026 + 0.1323i



2.2 – 2.0i

0.0265 – 0.7644i



C

3.0 – 2.0i

0.7000 – 0.9000i


3.0 – 1.5i

0.9262 – 1.3292i



3.0 – 1.0i

1.6000 – 1.8000i



3.0 – 0.5i

2.9765 – 1.6941i



3.0

4.0000



3.0 + 0.5i

2.9765 + 1.6941i



3.0 + 1.0i

1.6000 + 1.8000i



3.0 + 1.5i

0.9262 + 1.3292i



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 499311295