The open-loop transfer function of the system shown in the figure is:
\[
G(s) = \frac{K s (s + 2)}{(s + 5)(s + 7)}
\]
For \( K \geq 0 \), which of the following real-axis points is on the root locus?
Show Hint
- The root locus lies on the real axis between two poles or zeros.
- For this case, the root locus exists between \( s = -5 \) and \( s = -7 \).
- \( s = -2 \)
- \( s = -5 \)
- \( s = -7 \)
- \( s = 0 \)
The Correct Option is B
Solution and Explanation
The root locus of a system is a plot of the poles of the closed-loop transfer function as the system gain \( K \) varies from 0 to \( \infty \). The root locus is drawn along the real axis between poles and zeros, and the general rules for identifying points on the root locus are as follows:
- The root locus exists on the real axis between two poles or two zeros.
- The root locus starts at the poles of the open-loop transfer function and ends at the zeros.
Given the transfer function: \[ G(s) = \frac{K s (s + 2)}{(s + 5)(s + 7)} \] - The poles are at \( s = -5 \) and \( s = -7 \), and the zeros are at \( s = 0 \) and \( s = -2 \).
- The root locus exists along the real axis between the poles at \( s = -5 \) and \( s = -7 \), as this region is between two poles.
Thus, the real-axis point \( s = -5 \) is on the root locus.
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