Question

If the open loop transfer function of the negative feedback control system is \[ G(s)H(s) = \frac{k}{(s+1)^3}, \] then the gain $k$ for a closed loop pole at \[ \left( \frac{1}{2} + j\frac{\sqrt{3}}{2} \right) \] is:

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

To find gain $k$, substitute the closed-loop pole into the characteristic equation $1 + G(s)H(s) = 0$.

Answer 1

The Correct Option is A

Closed-loop poles satisfy: \[ 1 + G(s)H(s) = 0 \Rightarrow 1 + \frac{k}{(s+1)^3} = 0 \] Let $s = \frac{1}{2} + j\frac{\sqrt{3}}{2}$ \[ (s+1) = \left( \frac{3}{2} + j\frac{\sqrt{3}}{2} \right) \] \[ (s+1)^3 = \left( \frac{3}{2} + j\frac{\sqrt{3}}{2} \right)^3 = 1/k \Rightarrow k = 1 \]