Question

A system with transfer function:
$G\left(s\right)=\frac{(s^2+9)(s+2)}{(9s+1)(s+3)(s+4)}$
is excited $sin\left(\omega t\right)$. The steady-state output of the system is zero at

• A. $\omega =1rad/s$
• B. $\omega =2rad/s$
• C. $\omega =3rad/s$
• D. $\omega =4rad/s$

Answer 1

The correct option is C $\omega =3rad/s$

For sinusoidal excitation,
$s=j\omega$
$\therefore G\left(j\omega \right)=\frac{(-{\omega }^2+9)(j\omega +2)}{(j\omega +1)(j\omega +3)(j\omega +4)}$
For zero steady-state output
$|G\left(j\omega \right)|=0$
=$\frac{(-{\omega }^2+9)\sqrt{{\omega }^2+4}}{\left(\sqrt{{\omega }^2+1}\right)\left(\sqrt{{\omega }^2+9}\right)\left(\sqrt{{\omega }^2+16}\right)}$
For zero steady-state output,
$\Rightarrow {\omega }^2=9$
$\Rightarrow \omega =3rad/sec$