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

$e^{j2\pi }+\mathbf{\text{𝐆}}(s)=0$

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

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

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

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

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

Simplifying the characteristic equation,

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

from which we derive the angle condition:

$\pi +2k\pi ={\theta }_1+{\theta }_2+\cdots +{\theta }_n-({\phi }_1+{\phi }_2+\cdots +{\phi }_m)$

for $k=0,1,2,\dots$,

${\theta }_1,{\theta }_2,\dots ,{\theta }_n$

are the angles of zeros 1 to n, and

${\phi }_1,{\phi }_2,\dots ,{\phi }_m$

are the angles of poles 1 to m.

The magnitude condition is derived similarly.

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