Magnitude condition

The magnitude 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 angle 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, $A_{p}e^{j\theta _{p}}$ and $B_{q}e^{j\phi _{q}}$, respectively, ${\textbf {G}}(s)$ may be rewritten.

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

Simplifying the characteristic equation,

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

from which we derive the magnitude condition:

$1=K{\frac {A_{1}A_{2}\cdots A_{n}}{B_{1}B_{2}\cdots B_{m}}}.$

The angle condition is derived similarly.