Question

The open loop transfer function of a unity fedback system is given by $G\left(s\right)=\left(e^{-0.1s}\right)/s$. The gain margin of this system is

• A. 11.95 dB
• B. 17.67 dB
• C. 21.33 dB
• D. 23.9 dB

Answer 1

The correct option is D 23.9 dB

Open loop transfer function, $G\left(s\right)=\left(e^{-0.1s}\right)/s$.
Put, $s=j\omega$
$G\left(j\omega \right)=\frac{e^{-j0.1\omega }}{j\omega }$
At phase crossover frequency $\left({\omega }_{pc}\right)$ , phase of OLTF is $-{180}^{\circ }$
$\angle G\left(j\omega \right){|}_{\omega /{\omega }_{pc}}=-{180}^{\circ }=-\pi$
$-\frac{\pi }{2}-0.1{\omega }_{pc}=-\pi$
$-0.1{\omega }_{pc}=-\frac{\pi }{2}$
${\omega }_{pc}=5\pi$
|$e^{-0.1\omega }|$ is always 1 for any vlue of $\omega$
$|G\left(j\omega \right){|}_{{\omega }_{pc=5}}=\frac{1}{\omega }$
Gain at ${\omega }_{pc}$(phase-cross frequency)
$|G\left(j\omega \right){|}_{{\omega }_{pc=5}}=\frac{1}{5\pi }$
$Gainmargin=20log\frac{1}{|G\left(j{\omega }_{pc}\right)|}$
$=20log5\pi =23.9dB$