Question

An open loop transfer function of a unity negative feedback control system is given by,
$G\left(s\right)=\frac{K}{s(s+a)(s+b)}$
If the gain margin of the system is 32 dB for K = 2, then the value of K required to get the gain margin of 25 dB will be

• A. 4.48
• B. 38.1
• C. 1.12
• D. 6.38

Answer 1

The correct option is A 4.48

The gain margin of the system can be given as,

$GM=20{\log }_{10}\frac{1}{|G\left(j{\omega }_{pc}\right)|}$

${\omega }_{pc}$ is independent of the value of K.

So, $GM=C-20lo{g}_{10}\left(K\right)$

Where, C is a term independent of "K".

For K = 2, $GM=32dB$
So, $C=32+20lo{g}_{10}\left(2\right)$

When GM = 25 dB,
$25=C-20{\log }_{10}\left(K\right)$
$25=32+20{\log }_{10}\left(2\right)-20lo{g}_{10}\left(K\right)$
$20{\log }_{10}\left(K\right)=7+20{\log }_{10}\left(2\right)=13.02$

$K={10}^{\left(13.02/20\right)}=4.48$