- Angle condition
In mathematics, the angle condition is a constraint that is satisfied by the locus of points in the
s-plane on whichclosed-loop pole s of a system reside. In combination with themagnitude condition , these two mathematical expressions fully determine theroot locus .Let the
characteristic equation of a system be 1+ extbf{G}(s)=0, where extbf{G}(s)=frac{ extbf{P}(s)}{ extbf{Q}(s)}. Rewriting the equation inpolar form is useful.e^{j2pi}+ extbf{G}(s)=0
extbf{G}(s)=-1=e^{j(pi+2kpi)} wherek=0,1,2,...) are the only solutions to this equation. Rewriting extbf{G}(s) in factored form,
extbf{G}(s)=frac{ extbf{P}(s)}{ extbf{Q}(s)}=Kfrac{(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_pe^{j heta_p} and B_qe^{jphi_q}, respectively, extbf{G}(s) may be rewritten.
extbf{G}(s)=Kfrac{A_1 A_2 cdots A_ne^{j( heta_1+ heta_2+cdots+ heta_n){B_1 B_2 cdots B_m e^{j(phi_1+phi_2+cdots+phi_m)
Simplifying the characteristic equation,
e^{j(pi+2kpi)}=Kfrac{A_1 A_2 cdots A_ne^{j( heta_1+ heta_2+cdots+ heta_n){B_1 B_2 cdots B_m e^{j(phi_1+phi_2+cdots+phi_m)=Kfrac{A_1 A_2 cdots A_n}{B_1 B_2 cdots B_m}e^{j( heta_1+ heta_2+cdots+ heta_n-(phi_1+phi_2+cdots+phi_m))},
from which we derive the angle condition:
pi+2kpi= heta_1+ heta_2+cdots+ heta_n-(phi_1+phi_2+cdots+phi_m) wherek=0,1,2,...).
The
magnitude condition is derived similarly.
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