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| Ask |
For the control system in the given figure, the value of K for critical damping is
(a) 1
(b) 5.125
(c) 6.831
(d) 10
Answer is ( b )
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| Concept |
We should first consider simplifying the block - diagram in the question as:
Combining the two cascaded blocks , we get : $G(s) = (K)(\frac{2}{s^2+7s+2})$
A closed loop control system having a form as below:
Thus , the above diagram simplifies to $\frac{C(s)}{R(s)} = \frac{(\frac{2K}{s^2+7s+2})}{1+(\frac{2K}{s^2+7s+2})}$
$\implies \frac{C(s)}{R(s)}= \frac{2K}{s^2+7s+2+2K}$
Now to Solve the Above Equation we need to understand the characteristic equation of a second - order system.
In general a second - order control system equation is given by :
$$\frac{C(s)}{R(s)}= \frac{\omega_n^2}{s^2+2\zeta\omega_n s+\omega_n^2}$$
where,
$\zeta$ = damping ratio ; $\omega_n$ = natural frequency
The characteristic Equation of the system is given by :$s^2+2\zeta\omega_n s+\omega_n^2 = 0$
Comparing this equation to an algebraic equation of the form $ax^2+bx+c=0$
The roots of the above equation are $ x= \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
For a critical damping case $ b^2 - 4ac= 0 , \implies x= \frac{-b}{2a}$
The characteristic equation for the given problem : $s^2+7s+2+2K = 0 $
As for critical damping case $ b^2 - 4ac= 0 $
$\implies 7^2 - 4(1)(2+2K)= 0 $
$\implies K=41/8 = 5.125 $
(b) 5.125
