Question

The Root Locus always starts at the _______ and ends at the _______.

• A. Zeros; Poles
• B. Poles; Zeros
• C. Origin; Infinity
• D. Real axis; Imaginary axis

Hint:

Remember:
Start \( \implies K = 0 \implies \) Open-Loop Poles.
End \( \implies K = \infty \implies \) Open-Loop Zeros (finite or at infinity).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the starting and ending locations of the root locus branches in the s-plane.
The Root Locus is a graphical representation of the paths of the closed-loop poles of a system as the open-loop gain \( K \) varies from 0 to infinity.

Step 2: Key Formula or Approach:
The closed-loop transfer function of a unity feedback system is:
\[ T(s) = \frac{K G(s)}{1 + K G(s)} \] The closed-loop poles are the roots of the characteristic equation:
\[ 1 + K G(s) = 0 \implies G(s) = -\frac{1}{K} \] Let \( G(s) = \frac{N(s)}{D(s)} \), where the roots of \( D(s) = 0 \) are the open-loop poles and the roots of \( N(s) = 0 \) are the open-loop zeros.
Substituting this gives:
\[ D(s) + K N(s) = 0 \]

Step 3: Detailed Explanation:
Let us analyze the limits of the gain \( K \) to find the start and end points of the locus:

• Starting Points (\( K = 0 \)):
- When we set \( K = 0 \) in the characteristic equation \( D(s) + K N(s) = 0 \), we get:
\[ D(s) = 0 \] - This equation defines the open-loop poles of the system.
- Therefore, at \( K = 0 \), the closed-loop poles are identical to the open-loop poles. This means the root locus branches always start at the open-loop poles.

• Ending Points (\( K \to \infty \)):
- As \( K \) approaches infinity, we divide the characteristic equation by \( K \):
\[ \frac{D(s)}{K} + N(s) = 0 \implies N(s) = 0 \] - This equation defines the open-loop zeros of the system.
- Therefore, as \( K \to \infty \), the closed-loop poles approach the open-loop zeros.
- If the system has more poles (\( P \)) than zeros (\( Z \)), then \( P - Z \) branches will terminate at zeros located at infinity along asymptotic paths.

Step 4: Final Answer:
The root locus branches always start at the open-loop poles (for \( K = 0 \)) and end at the open-loop zeros (for \( K \to \infty \)).