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 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.

e^{j2\pi}+\textbf{G}(s)=0

\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,

\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)},

and representing each factor (s − ap) and (s − bq) 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:

\pi+2k\pi=\theta_1+\theta_2+\cdots+\theta_n-(\phi_1+\phi_2+\cdots+\phi_m) where (k = 0,1,2,...).

The magnitude condition is derived similarly.

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