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. 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 + G ( s ) = 0 {displaystyle 1+{ extbf {G}}(s)=0} , where G ( s ) = P ( s ) Q ( s ) {displaystyle { extbf {G}}(s)={frac {{ extbf {P}}(s)}{{ extbf {Q}}(s)}}} . Rewriting the equation in polar form is useful. G ( s ) = − 1 = e j ( π + 2 k π ) {displaystyle { extbf {G}}(s)=-1=e^{j(pi +2kpi )}} where ( k = 0 , 1 , 2 , . . . ) {displaystyle (k=0,1,2,...)} are the only solutions to this equation. Rewriting G ( s ) {displaystyle { extbf {G}}(s)} in factored form, and representing each factor ( s − a p ) {displaystyle (s-a_{p})} and ( s − b q ) {displaystyle (s-b_{q})} by their vector equivalents, A p e j θ p {displaystyle A_{p}e^{j heta _{p}}} and B q e j ϕ q {displaystyle B_{q}e^{jphi _{q}}} , respectively, G ( s ) {displaystyle { extbf {G}}(s)} may be rewritten.