# Angle condition

Angle condition

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 $1+ extbf\left\{G\right\}\left(s\right)=0$, where $extbf\left\{G\right\}\left(s\right)=frac\left\{ extbf\left\{P\right\}\left(s\right)\right\}\left\{ extbf\left\{Q\right\}\left(s\right)\right\}$. Rewriting the equation in polar form is useful.

$e^\left\{j2pi\right\}+ extbf\left\{G\right\}\left(s\right)=0$

$extbf\left\{G\right\}\left(s\right)=-1=e^\left\{j\left(pi+2kpi\right)\right\}$ where$\left(k=0,1,2,...\right)$ are the only solutions to this equation. Rewriting $extbf\left\{G\right\}\left(s\right)$ in factored form,

$extbf\left\{G\right\}\left(s\right)=frac\left\{ extbf\left\{P\right\}\left(s\right)\right\}\left\{ extbf\left\{Q\right\}\left(s\right)\right\}=Kfrac\left\{\left(s-a_1\right)\left(s-a_2\right)cdots\left(s-a_n\right)\right\}\left\{\left(s-b_1\right)\left(s-b_2\right)cdots\left(s-b_m\right)\right\}$,

and representing each factor $\left(s-a_p\right)$ and $\left(s-b_q\right)$ by their vector equivalents, $A_pe^\left\{j heta_p\right\}$ and $B_qe^\left\{jphi_q\right\}$, respectively, $extbf\left\{G\right\}\left(s\right)$ may be rewritten.

$extbf\left\{G\right\}\left(s\right)=Kfrac\left\{A_1 A_2 cdots A_ne^\left\{j\left( heta_1+ heta_2+cdots+ heta_n\right)\left\{B_1 B_2 cdots B_m e^\left\{j\left(phi_1+phi_2+cdots+phi_m\right)$

Simplifying the characteristic equation,

$e^\left\{j\left(pi+2kpi\right)\right\}=Kfrac\left\{A_1 A_2 cdots A_ne^\left\{j\left( heta_1+ heta_2+cdots+ heta_n\right)\left\{B_1 B_2 cdots B_m e^\left\{j\left(phi_1+phi_2+cdots+phi_m\right)=Kfrac\left\{A_1 A_2 cdots A_n\right\}\left\{B_1 B_2 cdots B_m\right\}e^\left\{j\left( heta_1+ heta_2+cdots+ heta_n-\left(phi_1+phi_2+cdots+phi_m\right)\right)\right\}$,

from which we derive the angle condition:

$pi+2kpi= heta_1+ heta_2+cdots+ heta_n-\left(phi_1+phi_2+cdots+phi_m\right)$ where$\left(k=0,1,2,...\right)$.

The magnitude condition is derived similarly.

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