The open loop transfer function of a unity feedback control system is given as \$G(s) = \frac{as+1}{s^2}\$. What value of 'a' will give a phase margin of 45° ?
\$G(s) = \frac{as+1}{s^2}\$
\$Transfer\$ \$function\$, \$T(s)=\frac{G(s)}{1+G(s)*1}\$
\$T(s)=\frac{as+1}{s^2+as+1}\$
\$T(s)=\frac{as+1}{(s+\frac{a}{2})^2+1-\frac{a^2}{4}}\$
\$T(s)=\frac{as+1}{(s+\frac{a}{2})^2+\left(\sqrt{1-\frac{a^2}{4}}\right)^2}\$
Let, \$\omega=\sqrt{1-\frac{a^2}{4}}\$
\$T(s)=\frac{as+1}{(s+\frac{a}{2})^2+{\omega}^2}\$
\$T(s)=\frac{a(s+\frac{a}{2})+(1-\frac{a^2}{2})}{(s+\frac{a}{2})^2+{\omega}^2}\$