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

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

$\text{G}(s)=-1=e^{j(\pi +2k\pi )}$ where $(k=0,1,2,...)$ are the only solutions to this equation. Rewriting $\text{G}(s)$ in factored form,

$\text{G}(s)=\frac{\text{P}(s)}{\text{Q}(s)}=K\frac{(s-a_1)(s-a_2)\cdots (s-a_n)}{(s-b_1)(s-b_2)\cdots (s-b_m)},$

and representing each factor $(s-a_p)$ and $(s-b_q)$ by their vector equivalents, $A_p{e}^{j{\theta }_p}$ and $B_q{e}^{j{\varphi }_q}$, respectively, $\text{G}(s)$ may be rewritten.

$\text{G}(s)=K\frac{A_1{A}_2\cdots A_n{e}^{j({\theta }_1+{\theta }_2+\cdots +{\theta }_n)}}{B_1{B}_2\cdots B_m{e}^{j({\varphi }_1+{\varphi }_2+\cdots +{\varphi }_m)}}$

Simplifying the characteristic equation,

$\begin{array}{rl}{e}^{j(\pi +2k\pi )}& =K\frac{A_1{A}_2\cdots A_n{e}^{j({\theta }_1+{\theta }_2+\cdots +{\theta }_n)}}{B_1{B}_2\cdots B_m{e}^{j({\varphi }_1+{\varphi }_2+\cdots +{\varphi }_m)}}\\ & =K\frac{A_1{A}_2\cdots A_n}{B_1{B}_2\cdots B_m}{e}^{j({\theta }_1+{\theta }_2+\cdots +{\theta }_n-({\varphi }_1+{\varphi }_2+\cdots +{\varphi }_m))},\end{array}$

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.