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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?
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Most Upvoted 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:
∠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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System is said to be marginally stable, ifa)Gain crossover frequency &...
Marginally Stable System Explanation:
Marginally stable systems are characterized by having the gain crossover frequency equal to the phase crossover frequency. This condition indicates that the system is on the brink of stability, with a phase margin of exactly 0 degrees.
Explanation of Option 'B':
- Gain crossover frequency = Phase crossover frequency: This condition signifies that the gain margin and phase margin are both zero, which is a defining characteristic of a marginally stable system.
Therefore, in a marginally stable system, the gain crossover frequency is equal to the phase crossover frequency. This delicate balance between gain and phase margins is what defines the system's marginally stable behavior.
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System is said to be marginally stable, ifa)Gain crossover frequency > Phase crossover frequencyb)Gain crossover frequency = Phase crossover frequencyc)Gain crossover frequency < Phase crossover frequencyd)Gain crossover frequency ≠ Phase crossover frequencyCorrect answer is option 'B'. Can you explain this answer?
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