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System is said to be marginally stable, if
- a)Gain crossover frequency > Phase crossover frequency
- b)Gain crossover frequency = Phase crossover frequency
- c)Gain crossover frequency < Phase crossover frequency
- d)Gain crossover frequency ≠ Phase crossover frequency
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
System is said to be marginally stable, ifa)Gain crossover frequency &...
Explanation:
Marginally Stable System:
A marginally stable system is a system where the gain crossover frequency is equal to the phase crossover frequency. In other words, the gain margin and phase margin are both zero.
Gain Crossover Frequency:
The gain crossover frequency is the frequency at which the magnitude of the open-loop transfer function is equal to 1 or 0 dB.
Phase Crossover Frequency:
The phase crossover frequency is the frequency at which the phase of the open-loop transfer function is equal to -180 degrees.
Relation between Gain and Phase Crossover Frequencies:
In a marginally stable system, the gain crossover frequency is equal to the phase crossover frequency. This means that the system is on the verge of instability, as there is no margin for stability. Any slight variation in parameters can potentially push the system into instability.
Conclusion:
Therefore, in a marginally stable system, the gain crossover frequency is equal to the phase crossover frequency. This balance between gain and phase margins indicates a critical stability point for the system.
Marginally Stable System:
A marginally stable system is a system where the gain crossover frequency is equal to the phase crossover frequency. In other words, the gain margin and phase margin are both zero.
Gain Crossover Frequency:
The gain crossover frequency is the frequency at which the magnitude of the open-loop transfer function is equal to 1 or 0 dB.
Phase Crossover Frequency:
The phase crossover frequency is the frequency at which the phase of the open-loop transfer function is equal to -180 degrees.
Relation between Gain and Phase Crossover Frequencies:
In a marginally stable system, the gain crossover frequency is equal to the phase crossover frequency. This means that the system is on the verge of instability, as there is no margin for stability. Any slight variation in parameters can potentially push the system into instability.
Conclusion:
Therefore, in a marginally stable system, the gain crossover frequency is equal to the phase crossover frequency. This balance between gain and phase margins indicates a critical stability point for the system.
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Community Answer
System is said to be marginally stable, ifa)Gain crossover frequency &...
Gain Margin:
- The gain margin is a factor by which the gain of a stable system is allowed to increase before the system reaches the verge of instability.
- It is also defined as the reciprocal of the gain at which the phase angle becomes 1800.
- The frequency at which the phase angle is 1800 is called the phase crossover frequency (ωpc).
Mathematically,
∠G(jω)H(jω) = -180º
If |G(jω)H(jω)| = a, at ω = ωpc
Then Gain margin,
Phase Margin:
- It is the amount of additional phase lag (at gain crossover frequency) that is required to bring the system to the verge of instability.
- The frequency at which |G(jω)H(jω|, the magnitude of the open-loop transfer function is unity is called gain crossover frequency (ωpc).
At ωpc, |G(jω)H(jω| = 1
if, ∠G(jω)H(jω) = ϕ at ω = ωgc
Then, P.M. = 180 + ϕ
Analysis:
For stable systems,
|G(jωpc)H(jωpc| < 1, and
∠G(jωgc)H(jωgc) > -180
so that, gain margin and phase margin both are positive
which is possible when ωgc < ωpc
For marginally stable systems,
ωgc = ωpc
Hence option (b) is the correct answer.
if, ∠G(jω)H(jω) = ϕ at ω = ωgc
Then, P.M. = 180 + ϕ
Analysis:
For stable systems,
|G(jωpc)H(jωpc| < 1, and
∠G(jωgc)H(jωgc) > -180
so that, gain margin and phase margin both are positive
which is possible when ωgc < ωpc
For marginally stable systems,
ωgc = ωpc
Hence option (b) is the correct answer.
Note: For stable systems having two or more gain crossover frequencies, the phase margin is measured at the highest gain crossover frequency.
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