Magnitude condition
The magnitude 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 angle 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 magnitude condition:
The angle condition is derived similarly.
