Consider the unity-negative-feedback system shown in Figure (i) below, where gain \( K \geq 0 \). The root locus of this system is shown in Figure (ii) below. For what value(s) of \( K \) will the system in Figure (i) have a pole at \( -1 + j1 \)?
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In root locus analysis, the path of the poles is determined by the locations of the open-loop poles and zeros. The root locus shows where the poles of the closed-loop system will move as \( K \) increases, but in some cases, certain complex points (like \( -1 + j1 \)) are never reached by the poles.
In control systems, the root locus provides a graphical method for determining how the poles of the closed-loop transfer function vary as the system gain \( K \) is varied. The root locus plot helps us analyze the stability and behavior of the system as \( K \) increases from \( 0 \) to \( \infty \).
Step 1: Understand the system and root locus plot
The system is a unity-negative-feedback system, meaning that the feedback is negative, and the transfer function is \( \frac{G(s)}{1 + G(s)} \) where \( G(s) \) is the open-loop transfer function.
The root locus plot, shown in Figure (ii), illustrates the movement of the poles of the closed-loop transfer function as \( K \) varies.
The key features of the root locus are:
1. The poles of the system are initially at specific locations in the complex plane.
2. As \( K \) increases, the poles move along specific paths (the root locus).
3. The root locus shows the paths along which the system poles move, and at which value of \( K \) they cross the imaginary axis or settle at specific points.
Step 2: Analyze the root locus plot (Figure ii)
From Figure (ii), we observe:
The poles of the open-loop system are initially located on the real axis, and as the gain \( K \) increases, the poles start to move along the real axis.
There is a circular movement in the root locus that is centered around specific points on the real axis.
Step 3: Locate the point \( -1 + j1 \) on the complex plane
We are asked to find the value of \( K \) at which the system has a pole at \( -1 + j1 \), which is a complex point in the left half of the complex plane.
\( -1 + j1 \) is located in the left-half plane, specifically 1 unit left of the real axis and 1 unit above the real axis on the imaginary axis.
To have a pole at this point, the root locus must pass through this point for some value of \( K \).
Step 4: Interpretation of the root locus plot
By carefully analyzing the root locus plot, we can see that the locus does not pass through the point \( -1 + j1 \) at any positive value of \( K \). The root locus plot indicates that the poles move along certain paths, but none of the paths intersect at \( -1 + j1 \) for any positive value of \( K \).
Thus, no positive value of \( K \) results in a pole exactly at \( -1 + j1 \).
Conclusion:
The correct answer is (C): For no positive value of \( K \), the system will have a pole at \( -1 + j1 \).
