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.

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