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 Angle condition ： ウィキペディア英語版
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 $1+\textbf\left(s\right)=0$, where $\textbf\left(s\right)=\frac$. Rewriting the equation in polar form is useful.
$e^+\textbf\left(s\right)=0$
$\textbf\left(s\right)=-1=e^$ where
$\left(k=0,1,2,...\right)$ are the only solutions to this equation. Rewriting $\textbf\left(s\right)$ in factored form,
$\textbf\left(s\right)=\frac=K\frac$,
and representing each factor $\left(s-a_p\right)$ and $\left(s-b_q\right)$ by their vector equivalents, $A_pe^$ and $B_qe^$, respectively, $\textbf\left(s\right)$ may be rewritten.
$\textbf\left(s\right)=K\frac\right\}$
Simplifying the characteristic equation,
$e^=K\frac\right\}=K\frace^$,
from which we derive the angle condition:
$\pi+2k\pi=\theta_1+\theta_2+\cdots+\theta_n-\left(\phi_1+\phi_2+\cdots+\phi_m\right)$
where $\left(k=0,1,2,...\right)$,
$\theta_1,\theta_2, \cdots \theta_n$ are the angles of poles 1 to n, and
$\phi_1,\phi_2, \cdots \phi_m$ are the angles of zeros 1 to m.
The magnitude condition is derived similarly.

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