Angle condition

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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[disambiguation needed] of a system be 1+\textbf{G}(s)=0, where \textbf{G}(s)=\frac{\textbf{P}(s)}{\textbf{Q}(s)}. Rewriting the equation in polar form is useful.


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


and representing each factor (s-a_p) and (s-b_q) by their vector equivalents, A_pe^{j\theta_p} and B_qe^{j\phi_q}, respectively, \textbf{G}(s) may be rewritten.

\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)}}

Simplifying the characteristic equation,

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))},

from which we derive the angle condition:


where (k=0,1,2,...),

\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.