Magnitude condition

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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.