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Assertion (A): From the root sensitivity standpoint, a system should not be operated at the breakaway points.
Reason (R): The root sensitivity at the breakaway points is zero.
Reason (R): The root sensitivity at the breakaway points is zero.
- a)Both A and R are true and R is a correct explanation of A.
- b)Both A and R are true but R is not a correct explanation of A.
- c)A is true but R is false.
- d)A is false but R is true.
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
Assertion (A): From the root sensitivity standpoint, a system should n...
From the root sensitivity, standpoint, a system should not be operated at1the breakaway points because root sensitivity at breakaway point is infinite.
i.e.
at breakaway points
Here, α = closed loop poles(s)
and β = system gain (K)
Thus,
or,
i.e.
at breakaway points
Here, α = closed loop poles(s)
and β = system gain (K)
Thus,
or,
Most Upvoted Answer
Assertion (A): From the root sensitivity standpoint, a system should n...
A. Assertion: From the root sensitivity standpoint, a system should not be operated at the breakaway points.
B. Reason: The root sensitivity at the breakaway points is zero.
Explanation:
Root Sensitivity:
Root sensitivity is a measure of how sensitive the closed-loop poles of a control system are to changes in the parameters or coefficients of the system. It quantifies the change in the location of the closed-loop poles with respect to changes in the system parameters.
Breakaway Points:
Breakaway points are the points in the complex plane where the root locus curve intersects the imaginary axis. At these points, the complex conjugate poles of the system merge and become real poles, resulting in a change in the system's stability.
Understanding the Assertion:
The assertion states that a system should not be operated at the breakaway points, considering the root sensitivity standpoint. This means that operating the system at the breakaway points can lead to undesirable consequences due to high sensitivity to parameter changes.
Understanding the Reason:
The reason provided states that the root sensitivity at the breakaway points is zero. This implies that the closed-loop poles, which determine the system's stability, do not change with variations in the system parameters at the breakaway points.
Analysis:
While both the assertion and reason are true, they do not complement each other. The reason does not explain why the system should not be operated at the breakaway points from the root sensitivity standpoint.
Conclusion:
Hence, the correct answer is option 'C' - A is true but R is false. The system should not be operated at the breakaway points due to potential instability and not because the root sensitivity is zero at those points.
B. Reason: The root sensitivity at the breakaway points is zero.
Explanation:
Root Sensitivity:
Root sensitivity is a measure of how sensitive the closed-loop poles of a control system are to changes in the parameters or coefficients of the system. It quantifies the change in the location of the closed-loop poles with respect to changes in the system parameters.
Breakaway Points:
Breakaway points are the points in the complex plane where the root locus curve intersects the imaginary axis. At these points, the complex conjugate poles of the system merge and become real poles, resulting in a change in the system's stability.
Understanding the Assertion:
The assertion states that a system should not be operated at the breakaway points, considering the root sensitivity standpoint. This means that operating the system at the breakaway points can lead to undesirable consequences due to high sensitivity to parameter changes.
Understanding the Reason:
The reason provided states that the root sensitivity at the breakaway points is zero. This implies that the closed-loop poles, which determine the system's stability, do not change with variations in the system parameters at the breakaway points.
Analysis:
While both the assertion and reason are true, they do not complement each other. The reason does not explain why the system should not be operated at the breakaway points from the root sensitivity standpoint.
Conclusion:
Hence, the correct answer is option 'C' - A is true but R is false. The system should not be operated at the breakaway points due to potential instability and not because the root sensitivity is zero at those points.
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