Angle condition

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In mathematics, the angle condition is a constraint that is satisfied by the locus of points in the s-plane on which closed-loop poles of a system reside. In combination with the magnitude condition, these two mathematical expressions fully determine the root locus.

Let the characteristic equation of a system be , where . Rewriting the equation in polar form is useful.

where are the only solutions to this equation. Rewriting in factored form,

and representing each factor and by their vector equivalents, and , respectively, may be rewritten.

Simplifying the characteristic equation,

from which we derive the angle condition:

for ,

are the angles of poles 1 to n, and

are the angles of zeros 1 to m.

The magnitude condition is derived similarly.