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 $1+\textbf{G}(s)=0$, 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 $(s-a_p)$ 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:

$\pi+2k\pi=\theta_1+\theta_2+\cdots+\theta_n-(\phi_1+\phi_2+\cdots+\phi_m)$

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.