Question

A unity feedback system has the following open loop frequency response :

$\omega \left(rad/sec\right)$	2	3	4	5	6	8	10
|G$j\omega$|	7.5	4.8	3.15	2.25	1.70	1.00	0.64
$\angle$$(j\omega$)	$-{118}^o$	$-{130}^o$	$-{140}^o$	$-{150}^o$	$-{157}^o$	$-{170}^o$	$-{180}^o$

The gain and phase margin of the system are

• A. $0dB,-{180}^o$
• B. $3.88dB,-{170}^o$
• C. $0dB,{10}^o$
• D. $3.88dB,{10}^o$

Answer 1

The correct option is D $3.88dB,{10}^o$

Gain margin : Gain margin are defined at the point of phase cross-over frequency that is the gain when the phase crosses $-{180}^o$.
Phase margin are defined at the frequency when the gain becomes 1.0 dB.

$Gainmargin\left(GM\right)=20log\frac{1}{|G\left(j{\omega }_{\omega ={\omega }_p}|\right)|}$

Where,
${\omega }_p=Phasecross-overfrequency$
$=20log\frac{1}{0.64}=3.68dB.$
$Phasemargin{P}_m={180}^o+\angle G\left(j\omega \right){|}_{\omega ={\omega }_g}$
$Hence,P_m={180}^o-{170}^o={10}^o$