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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).
Most Upvoted Answer
What will be the nature of time response if the roots of the character...
The nature of the time response of a system is determined by the location of the roots of its characteristic equation in the s-plane. The s-plane is a complex plane where the real part of the roots represents the damping of the response and the imaginary part represents the frequency of oscillations.
When the roots of the characteristic equation are located on the imaginary axis of the s-plane, it means that the system has purely imaginary poles. In this case, the nature of the time response is given by option C, which states that there will be no oscillations.
Here's an explanation of why this is the case:
1. Purely imaginary poles:
When the roots of the characteristic equation are located on the imaginary axis, it means that the poles of the system have no real part. This implies that there is no damping in the system, as damping is represented by the real part of the poles.
2. No oscillations:
In a system with no damping, there are no oscillations. Oscillations occur when there is a balance between the energy stored in the system and the energy dissipated due to damping. In this case, since there is no damping, there is no dissipation of energy and hence no oscillations.
3. Time response:
The time response of a system with purely imaginary poles will be a sinusoidal function with a constant amplitude and frequency. However, since there is no damping, this sinusoidal response will continue indefinitely without decaying or growing.
To summarize, when the roots of the characteristic equation are located on the imaginary axis of the s-plane, the system has purely imaginary poles, indicating no damping. As a result, there will be no oscillations in the time response of the system.
When the roots of the characteristic equation are located on the imaginary axis of the s-plane, it means that the system has purely imaginary poles. In this case, the nature of the time response is given by option C, which states that there will be no oscillations.
Here's an explanation of why this is the case:
1. Purely imaginary poles:
When the roots of the characteristic equation are located on the imaginary axis, it means that the poles of the system have no real part. This implies that there is no damping in the system, as damping is represented by the real part of the poles.
2. No oscillations:
In a system with no damping, there are no oscillations. Oscillations occur when there is a balance between the energy stored in the system and the energy dissipated due to damping. In this case, since there is no damping, there is no dissipation of energy and hence no oscillations.
3. Time response:
The time response of a system with purely imaginary poles will be a sinusoidal function with a constant amplitude and frequency. However, since there is no damping, this sinusoidal response will continue indefinitely without decaying or growing.
To summarize, when the roots of the characteristic equation are located on the imaginary axis of the s-plane, the system has purely imaginary poles, indicating no damping. As a result, there will be no oscillations in the time response of the system.
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