Angle condition
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 zeros 1 to n, and
are the angles of poles 1 to m.
The magnitude condition is derived similarly.
