Question

The root-locus plot of a closed-loop system with unity negative feedback and transfer function $KG(s)$ in the forward path is shown in the figure. Note that $K$ is varied from 0 to $\infty$. Select the transfer function $G(s)$ that results in the shown root-locus.

• A. $G(s) = \dfrac{1}{(s + 1)^5}$
• B. $G(s) = \dfrac{1}{s^5 + 1}$
• C. $G(s) = \dfrac{s - 1}{(s + 1)^6}$
• D. $G(s) = \dfrac{s + 1}{s^6 + 1}$

Hint:

If a root-locus shows $n$ symmetrically spaced branches around a pole, the system has an $n$-th order repeated pole at that point.

Answer 1

The Correct Option is A

The figure shows five root-locus branches emerging from a single real pole located at $s = -1$. The angles between adjacent branches are exactly: \[ \frac{360^\circ}{5} = 72^\circ, \] which matches the diagram showing rays at $36^\circ$, $108^\circ$, $180^\circ$, $252^\circ$, and $324^\circ$ relative to the real axis. This pattern is characteristic of a 5th-order real pole, i.e., a pole of multiplicity 5. Thus, the forward path transfer function must have the form: \[ G(s) = \frac{1}{(s + 1)^5}, \] so that the characteristic equation \[ 1 + K G(s) = 0 \] produces 5 equally spaced root-locus angles around $s = -1$. None of the other options give a 5th-order pole at $s = -1$ or match the 72° symmetry. Hence, the correct choice is option (A). Final Answer: $\dfrac{1}{(s + 1)^5}$