Question

Figures shows a feedback system where k > 0



The range of k for which is stable will be given by

• A. 0 < k < 30
• B. 0 < k < 39
• C. 0 < k < 390
• D. k > 390

Answer 1

The correct option is C 0 < k < 390

G(s) = $\frac{k}{s(s+3)(s+10)}$

and H(s) = 1

Characteristic equation,

$\Rightarrow 1+G\left(s\right)H\left(s\right)=0$

$\Rightarrow 1+\frac{k}{s(s+3)(s+10)}=0$

$\Rightarrow s(s+3)(s+10)+k=0$

$\Rightarrow s^3+13s^2+30s+k=0$

Routh-Array



According to Routh-Hurwitz criterion.

For a stable system, signs of first column do not change

k > 0

and

$\frac{13\times 30-k}{13}>0$

Therefore system to be stable 0 < k < 390.