Question

A control system with damping ratio $\xi =\sqrt{3}$ is represented by the block diagram which employs proportional plus error rate control. The value of error rate constant K when unit step is given as input to system will be

• A. $\frac{1}{3}$
• B. $\frac{1}{2}$
• C. $\frac{1}{4}$
• D. $\frac{2}{3}$

Answer 1

The correct option is D $\frac{2}{3}$

The closed-loop transfer function for given system,

$\frac{G\left(s\right)}{1+G\left(s\right)}=\frac{\frac{3(1+sK)}{s(s+4)}}{1+\frac{3(1+sK)}{s(s+4)}}=\frac{3(1+sK)}{s(s+4)+3(1+sK)}$

$=\frac{3(1+sK)}{s^2+4s+3(1+sK)}=\frac{3+3sK}{s^2+(4+3K)s+3}$

Comparing the transfer function with standard second order transfer function.

${\omega }_n^2=3$

$\Rightarrow {\omega }_n=\sqrt{3}rad/\sec$

$2\xi {\omega }_n=4+3K$

$2\sqrt{3}\xi =3K+4$

$\frac{2\sqrt{3}\xi -4}{3}=K$

Given, $\xi =\sqrt{3},K=\frac{6-4}{3}=\frac{2}{3}$