Question

Consider the unity feedback control system shown. The value of K that results in phase margin of the system to be ${30}^{\circ }$ is
(Give the answer up to two decimal places.)

• A. 1.047

Answer 1

The correct option is A 1.047
Forward path transfer function,
$G\left(s\right)=\frac{K{e}^{-s}}{s}$

Given,
$Phasemargin={30}^{\circ }$

$orPhasemargin={180}^{\circ }+\varphi$

${30}^{\circ }={180}^{\circ }+\varphi$

$\varphi =-{150}^{\circ }$

$\varphi =\angle G(j\omega {)}_{\omega ={\omega }_{gc}}$

[where, ${\omega }_{gc}$ is gain crossover frequency]

$\angle G\left(j\omega \right)=-{90}^{\circ }-57.3{\omega }^{\circ }$

$At\omega ={\omega }_{gc}$
[gain crossover frequency]

$|G\left(j\omega \right)|=1$

$or\frac{K\times 1}{\omega }=1\Rightarrow \omega ={\omega }_{gc}=Krad/sec$

$\therefore \angle G(j\omega {)}_{\omega ={\omega }_{gc}}=-{90}^{\circ }-57.3K$

$-{90}^{\circ }-57.3K=-{150}^{\circ }$

$-57.3K=-{60}^{\circ }$

$K=\frac{60}{57.3}=1.047$