Special case when G(s)H(s) involves poles and/or zeros on the jωaxis

 

 

Fig 3: s-Plane contour and the G(s)H(s) locus in the GH plane

 

As seen from Figure 2, points D, E, and F on the semicircle of infinite radius in the s plane map into the origin of the GH plane. Since there is no pole in the right-half s plane and the G(s)H(s) locus does not encircle the -1 j0 point, there are no zeros of the function 1 G(s)H(s) in the right-half s plane. Therefore, the system is stable.

For an open-loop transfer function G(s)H(s) involving a 1/sⁿ factor (where n = 2,3, ...), the plot of G(s)H(s) has n clockwise semicircles of infinite radius about the origin as a representative point s moves along the semicircle of radius ε (where ε<< 1).

For example, consider the following open-loop transfer function:

 

As θ varies from -90° to 90° in the s plane, the angle of G(s)H(s) varies from 180° to -180°,as shown in Figure 3. Since there is no pole in the right-half s plane and the locus encircles the -1 j0 point twice clockwise for any positive value of K, there are two zeros of 1 G(s)H(s) in the right-half s plane. Therefore, this system is always unstable.