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: Assessment of Statement A.Breakaway and break-in points signify locations with multiple characteristic equation roots. While typically found on the real axis between two poles or zeros, they can exist in the complex plane if the poles/zeros originating the loci are complex. Thus, their location is not restricted to the real axis. Statement A is valid.
Step 2: Assessment of Statement B.By definition, the root locus represents all possible characteristic equation roots as gain K changes. Breakaway points are locations on these paths where roots separate from the real axis (or each other). Consequently, they must be situated on the root loci. Statement B is valid.
Step 3: Assessment of Statement C.Breakaway point locations are found by solving \(dK/ds = 0\), dependent on system pole and zero locations. For poles at -2 and -4, the breakaway is at -3. For poles at 0 and -1, it's at -0.5. No rule mandates placement between 0 and -1. Statement C is invalid.
Conclusion: Statements A and B are valid.