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{{Unreferenced|date=December 2009}}
In mathematics, the '''angle condition''' is a constraint that is satisfied by the locus of points in the [[s-plane]] on which [[closed-loop pole]]s of a system reside.  In combination with the [[magnitude condition]], these two mathematical expressions fully determine the [[root locus]].

Let the [[characteristic equation]]{{dn|date=December 2013}} of a system be <math>1+\textbf{G}(s)=0</math>, where <math>\textbf{G}(s)=\frac{\textbf{P}(s)}{\textbf{Q}(s)}</math>.  Rewriting the equation in [[polar form]] is useful.

<math>e^{j2\pi}+\textbf{G}(s)=0 </math>

<math>\textbf{G}(s)=-1=e^{j(\pi+2k\pi)}</math>  where
<math>(k=0,1,2,...)</math> are the only solutions to this equation.  Rewriting <math>\textbf{G}(s)</math> in [[factorization|factored form]],

<math>\textbf{G}(s)=\frac{\textbf{P}(s)}{\textbf{Q}(s)}=K\frac{(s-a_1)(s-a_2)\cdots(s-a_n)}{(s-b_1)(s-b_2)\cdots(s-b_m)}</math>,

and representing each factor <math>(s-a_p)</math> and <math>(s-b_q)</math> by their [[vector (geometric)|vector]] equivalents, <math>A_pe^{j\theta_p}</math> and <math>B_qe^{j\phi_q}</math>, respectively, <math>\textbf{G}(s)</math> may be rewritten.

<math>\textbf{G}(s)=K\frac{A_1 A_2 \cdots A_ne^{j(\theta_1+\theta_2+\cdots+\theta_n)}}{B_1 B_2 \cdots B_m e^{j(\phi_1+\phi_2+\cdots+\phi_m)}}</math>

Simplifying the characteristic equation, 

<math>e^{j(\pi+2k\pi)}=K\frac{A_1 A_2 \cdots A_ne^{j(\theta_1+\theta_2+\cdots+\theta_n)}}{B_1 B_2 \cdots B_m e^{j(\phi_1+\phi_2+\cdots+\phi_m)}}=K\frac{A_1 A_2 \cdots A_n}{B_1 B_2 \cdots B_m}e^{j(\theta_1+\theta_2+\cdots+\theta_n-(\phi_1+\phi_2+\cdots+\phi_m))}</math>,

from which we derive the angle condition:

<math>\pi+2k\pi=\theta_1+\theta_2+\cdots+\theta_n-(\phi_1+\phi_2+\cdots+\phi_m) </math>

where <math>(k=0,1,2,...)</math>, 

<math>\theta_1,\theta_2, \cdots \theta_n </math> are the angles of poles 1 to n, and

<math>\phi_1,\phi_2, \cdots \phi_m </math> are the angles of zeros 1 to m.

The [[magnitude condition]] is derived similarly.

{{DEFAULTSORT:Angle Condition}}
[[Category:Control theory]]