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

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

${\displaystyle {\textbf {G}}(s)=-1=e^{j(\pi +2k\pi )}}$

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

${\displaystyle {\textbf {G}}(s)={\frac {{\textbf {P}}(s)}{{\textbf {Q}}(s)}}=K{\frac {(s-a_{1})(s-a_{2})\cdots (s-a_{n})}{(s-b_{1})(s-b_{2})\cdots (s-b_{m})}},}$

and representing each factor ${\displaystyle (s-a_{p})}$ and ${\displaystyle (s-b_{q})}$ by their vector equivalents, ${\displaystyle A_{p}e^{j\theta _{p}}}$ and ${\displaystyle B_{q}e^{j\varphi _{q}}}$, respectively, ${\displaystyle {\textbf {G}}(s)}$ may be rewritten.

${\displaystyle {\textbf {G}}(s)=K{\frac {A_{1}A_{2}\cdots A_{n}e^{j(\theta _{1}+\theta _{2}+\cdots +\theta _{n})}}{B_{1}B_{2}\cdots B_{m}e^{j(\varphi _{1}+\varphi _{2}+\cdots +\varphi _{m})}}}}$

Simplifying the characteristic equation,

${\displaystyle {\begin{aligned}e^{j(\pi +2k\pi )}&=K{\frac {A_{1}A_{2}\cdots A_{n}e^{j(\theta _{1}+\theta _{2}+\cdots +\theta _{n})}}{B_{1}B_{2}\cdots B_{m}e^{j(\varphi _{1}+\varphi _{2}+\cdots +\varphi _{m})}}}\\[6pt]&=K{\frac {A_{1}A_{2}\cdots A_{n}}{B_{1}B_{2}\cdots B_{m}}}e^{j(\theta _{1}+\theta _{2}+\cdots +\theta _{n}-(\varphi _{1}+\varphi _{2}+\cdots +\varphi _{m}))},\end{aligned}}}$

from which we derive the angle condition:

${\displaystyle \pi +2k\pi =\theta _{1}+\theta _{2}+\cdots +\theta _{n}-(\varphi _{1}+\varphi _{2}+\cdots +\varphi _{m})}$

for ${\displaystyle k=0,1,2,\ldots }$,

${\displaystyle \theta _{1},\theta _{2},\ldots ,\theta _{n}}$

are the angles of zeros 1 to n, and

${\displaystyle \varphi _{1},\varphi _{2},\ldots ,\varphi _{m}}$

are the angles of poles 1 to m.

The magnitude condition is derived similarly.