Question

If the loop gain k of a negative feedback system having a loop transfer function k(s+3)/(s+8)${}^2$ is to be adjusted to induce a sustained oscillation then

• A. the frequency of this oscillation must be $4/\sqrt{3}rad/s$.
• B. the frequency of this oscillation must be 4 rad/s.
• C. the frequency of this oscillation must be 4 or $4/\sqrt{3}rad/s$.
• D. such a k does not exist.

Answer 1

The correct option is B the frequency of this oscillation must be 4 rad/s.

Open loop transfer function,

$G\left(s\right)H\left(s\right)=k\frac{(s+3)}{(s+8{)}^2}$

$\therefore$ Characteristic equation,

1 + G(s) H(s) = 0

$\Rightarrow 1+k\frac{(s+3)}{(s+8{)}^2}=0$

$\Rightarrow s^2+16s+64+k(s+3)=0$

$\Rightarrow s^2+(16+k)s+(64+3k)=0$

Routh-Array,



For sustained oscillation,

16 + k = 0

$\Rightarrow k=-16$

Auxiliary equation,

$s^2+64+3k=0$

$\Rightarrow s^2+64+3\times (-16)=0$

$\Rightarrow s^2+16=0$

$\Rightarrow s=\pm j4$

So, frequency of oscillation is 4 rad/sec.