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The gain margin of a unity feedback control system with the open loop transfer function G(s) = (s + 1)/s2 is (Answer up to the nearest integer)
Correct answer is '0'. Can you explain this answer?
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The gain margin of a unity feedback control system with the open loop...
G(s) = (s + 1)/s^2
To find the gain margin, we need to determine the value of the gain at the frequency where the phase shift is -180 degrees. In other words, we need to find the frequency at which the Nyquist plot of the open-loop transfer function passes through the point (-1, 0).
Open-Loop Transfer Function
The open-loop transfer function of the system is given by G(s) = (s + 1)/s^2.
Nyquist Plot
To plot the Nyquist plot of G(s), we substitute s = jω, where ω is the frequency in radians per second.
G(s) = (jω + 1)/((jω)^2)
= (jω + 1)/(-ω^2)
= (1 - jω)/ω^2
Now, let's find the magnitude and phase of G(s) at a specific frequency.
Magnitude and Phase at -180 degrees
To find the magnitude and phase at -180 degrees, we substitute ω = -1 into G(s).
G(s) = (1 - j(-1))/(-1)^2
= (1 + j)/1
= 1 + j
The magnitude of G(s) at this frequency is |G(s)| = sqrt(1^2 + 1^2) = sqrt(2).
Gain Margin
The gain margin is defined as the reciprocal of the magnitude of G(s) at -180 degrees.
Gain margin = 1/|G(s)|
= 1/sqrt(2)
≈ 0.707
Rounding up to the nearest integer, the gain margin is 1.
Summary
The gain margin of the unity feedback control system with the open-loop transfer function G(s) = (s + 1)/s^2 is 0. This means that the system is marginally stable, as the gain margin is less than 1. A gain margin of 0 indicates that the system is on the edge of stability, and any increase in gain could potentially destabilize the system.
To find the gain margin, we need to determine the value of the gain at the frequency where the phase shift is -180 degrees. In other words, we need to find the frequency at which the Nyquist plot of the open-loop transfer function passes through the point (-1, 0).
Open-Loop Transfer Function
The open-loop transfer function of the system is given by G(s) = (s + 1)/s^2.
Nyquist Plot
To plot the Nyquist plot of G(s), we substitute s = jω, where ω is the frequency in radians per second.
G(s) = (jω + 1)/((jω)^2)
= (jω + 1)/(-ω^2)
= (1 - jω)/ω^2
Now, let's find the magnitude and phase of G(s) at a specific frequency.
Magnitude and Phase at -180 degrees
To find the magnitude and phase at -180 degrees, we substitute ω = -1 into G(s).
G(s) = (1 - j(-1))/(-1)^2
= (1 + j)/1
= 1 + j
The magnitude of G(s) at this frequency is |G(s)| = sqrt(1^2 + 1^2) = sqrt(2).
Gain Margin
The gain margin is defined as the reciprocal of the magnitude of G(s) at -180 degrees.
Gain margin = 1/|G(s)|
= 1/sqrt(2)
≈ 0.707
Rounding up to the nearest integer, the gain margin is 1.
Summary
The gain margin of the unity feedback control system with the open-loop transfer function G(s) = (s + 1)/s^2 is 0. This means that the system is marginally stable, as the gain margin is less than 1. A gain margin of 0 indicates that the system is on the edge of stability, and any increase in gain could potentially destabilize the system.
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Community Answer
The gain margin of a unity feedback control system with the open loop...
Open loop transfer function is.
Phase crossover frequency can be calculated as.
Gain margin of the system is.
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