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