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

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

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

$extbf\{G\}(s)=frac\{\; extbf\{P\}(s)\}\{\; extbf\{Q\}(s)\}=Kfrac\{(s-a\_1)(s-a\_2)cdots(s-a\_n)\}\{(s-b\_1)(s-b\_2)cdots(s-b\_m)\}$,

and representing each factor $(s-a\_p)$ and $(s-b\_q)$ by their vector equivalents, $A\_pe^\{j\; heta\_p\}$ and $B\_qe^\{jphi\_q\}$, respectively, $extbf\{G\}(s)$ may be rewritten.

$extbf\{G\}(s)=Kfrac\{A\_1\; A\_2\; cdots\; A\_ne^\{j(\; heta\_1+\; heta\_2+cdots+\; heta\_n)\{B\_1\; B\_2\; cdots\; B\_m\; e^\{j(phi\_1+phi\_2+cdots+phi\_m)$

Simplifying the characteristic equation,

$e^\{j(pi+2kpi)\}=Kfrac\{A\_1\; A\_2\; cdots\; A\_ne^\{j(\; heta\_1+\; heta\_2+cdots+\; heta\_n)\{B\_1\; B\_2\; cdots\; B\_m\; e^\{j(phi\_1+phi\_2+cdots+phi\_m)=Kfrac\{A\_1\; A\_2\; cdots\; A\_n\}\{B\_1\; B\_2\; cdots\; B\_m\}e^\{j(\; heta\_1+\; heta\_2+cdots+\; heta\_n-(phi\_1+phi\_2+cdots+phi\_m))\}$,

from which we derive the angle condition:

$pi+2kpi=\; heta\_1+\; heta\_2+cdots+\; heta\_n-(phi\_1+phi\_2+cdots+phi\_m)$ where$(k=0,1,2,...)$.

The magnitude condition is derived similarly.