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What will be the nature of time response if the roots of the characteristic equation are located on the s-plane imaginary axis?
- a)Oscillations
- b)Damped oscillations
- c)No oscillations
- d)Under damped oscilaations
Correct answer is option 'C'. Can you explain this answer?
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What will be the nature of time response if the roots of the character...
Answer: c
Explanation: complex conjugate (non-multiple): oscillatory (sustained oscillations)
Complex conjugate (multiple): unstable (growing oscillations).
Explanation: complex conjugate (non-multiple): oscillatory (sustained oscillations)
Complex conjugate (multiple): unstable (growing oscillations).
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What will be the nature of time response if the roots of the character...
Nature of Time Response with Imaginary Axis Roots
When the roots of the characteristic equation are located on the s-plane imaginary axis, the nature of time response is no oscillations. This can be explained with the following points:
Definition of Imaginary Axis Roots:
In control system analysis, the characteristic equation represents the behavior of the closed-loop system. The roots of the characteristic equation are the values of s that satisfy the equation. When the roots are located on the imaginary axis, it means that the closed-loop system has complex conjugate poles with zero real part.
No Oscillations:
In a closed-loop system, the time response is the behavior of the system as it settles to its steady-state value. When the roots of the characteristic equation are on the imaginary axis, it means that there is no damping in the system. As a result, the system does not oscillate but instead settles to its steady-state value without any overshoot or oscillations.
Example:
For example, consider a closed-loop system with the characteristic equation s^2 + 4 = 0. The roots of this equation are s = ±2j. Since the roots are on the imaginary axis, the system does not oscillate but instead settles to its steady-state value without any overshoot or oscillations.
Conclusion:
In conclusion, when the roots of the characteristic equation are located on the s-plane imaginary axis, the nature of time response is no oscillations. This is because there is no damping in the system, and the system settles to its steady-state value without any overshoot or oscillations.
When the roots of the characteristic equation are located on the s-plane imaginary axis, the nature of time response is no oscillations. This can be explained with the following points:
Definition of Imaginary Axis Roots:
In control system analysis, the characteristic equation represents the behavior of the closed-loop system. The roots of the characteristic equation are the values of s that satisfy the equation. When the roots are located on the imaginary axis, it means that the closed-loop system has complex conjugate poles with zero real part.
No Oscillations:
In a closed-loop system, the time response is the behavior of the system as it settles to its steady-state value. When the roots of the characteristic equation are on the imaginary axis, it means that there is no damping in the system. As a result, the system does not oscillate but instead settles to its steady-state value without any overshoot or oscillations.
Example:
For example, consider a closed-loop system with the characteristic equation s^2 + 4 = 0. The roots of this equation are s = ±2j. Since the roots are on the imaginary axis, the system does not oscillate but instead settles to its steady-state value without any overshoot or oscillations.
Conclusion:
In conclusion, when the roots of the characteristic equation are located on the s-plane imaginary axis, the nature of time response is no oscillations. This is because there is no damping in the system, and the system settles to its steady-state value without any overshoot or oscillations.
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