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