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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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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.
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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