Question

In the root locus plot, the breakaway points
A. Need not always be on the real axis alone.
B. Must lie on the root loci.
C. Must lie between 0 and -1.
Choose the correct answer from the options given below:

• A. A, B and C only
• B. A and B only
• C. A and C only
• D. B and C only

Hint:

Remember the key rules of root locus: - Loci are the paths of the closed-loop poles. - Breakaway/break-in points are where loci leave or enter the real axis. They are always part of the locus. - Their location depends entirely on the open-loop pole/zero configuration.

Answer 1

The Correct Option is B

Step 1: Analyze Statement A. Breakaway (and break-in) points are locations where multiple roots of the characteristic equation are located. While they most commonly occur on the real axis between two poles or two zeros, they can exist in the complex plane if the poles/zeros from which the loci depart are complex. Therefore, they need not always be on the real axis alone. Statement A is correct.
Step 2: Analyze Statement B. The root locus is, by definition, the path of all possible roots of the characteristic equation as the gain K varies. Breakaway points are points on these paths where roots depart from the real axis (or each other). Therefore, they must lie on the root loci. Statement B is correct.
Step 3: Analyze Statement C. The location of breakaway points is determined by solving \(dK/ds = 0\), and they depend entirely on the location of the system's poles and zeros. For a system with poles at -2 and -4, the breakaway point is at -3. For a system with poles at 0 and -1, the point is at -0.5. There is no rule that they must lie between 0 and -1. Statement C is incorrect.
Conclusion: Statements A and B are correct.