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 [math]\displaystyle{ 1+\textbf{G}(s)=0 }[/math], where [math]\displaystyle{ \textbf{G}(s)=\frac{\textbf{P}(s)}{\textbf{Q}(s)} }[/math]. Rewriting the equation in polar form is useful.

[math]\displaystyle{ e^{j2\pi}+\textbf{G}(s)=0 }[/math]

[math]\displaystyle{ \textbf{G}(s)=-1=e^{j(\pi+2k\pi)} }[/math]

where [math]\displaystyle{ k=0,1,2,\ldots }[/math] are the only solutions to this equation. Rewriting [math]\displaystyle{ \textbf{G}(s) }[/math] in factored form,

[math]\displaystyle{ \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)}, }[/math]

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

[math]\displaystyle{ \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(\varphi_1+\varphi_2+\cdots+\varphi_m)}} }[/math]

Simplifying the characteristic equation,

[math]\displaystyle{ \begin{align} 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(\varphi_1+\varphi_2+\cdots+\varphi_m)}} \\[6pt] & = 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-(\varphi_1+\varphi_2+\cdots+\varphi_m))}, \end{align} }[/math]

from which we derive the angle condition:

[math]\displaystyle{ \pi+2k\pi=\theta_1+\theta_2+\cdots+\theta_n-(\varphi_1+\varphi_2+\cdots+\varphi_m) }[/math]

for [math]\displaystyle{ k=0,1,2,\ldots }[/math],

[math]\displaystyle{ \theta_1,\theta_2, \ldots, \theta_n }[/math]

are the angles of zeros 1 to n, and

[math]\displaystyle{ \varphi_1,\varphi_2, \ldots, \varphi_m }[/math]

are the angles of poles 1 to m.

The magnitude condition is derived similarly.