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

Root locus

Root locus analysis is a graphical method for examining how the roots of a system change with variation of a certain system parameter, commonly the gain of a feedback system. This is a technique used in the field of control systems developed by Walter R...

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Let the characteristic equation

Characteristic equation

Characteristic equation may refer to:* Characteristic equation , used to solve linear differential equations* Characteristic equation, a characteristic polynomial equation in linear algebra used to find eigenvalues...

 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:

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The angle condition is derived similarly.