The Nyquist plot of a stable open-loop system \( G(j\omega) \) is plotted in the frequency range \( 0 \leq \omega>\infty \) as shown. It is found to intersect a unit circle with center at the origin at the point \( P = -0.77 - 0.64j \). The points \( Q \) and \( R \) lie on \( G(j\omega) \) and assume values \( Q = 14.40 + 0.00j \) and \( R = -0.21 + 0.00j \). The phase margin (PM) and the gain margin (GM) of the system are _________
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The phase margin (PM) indicates the stability of the system in terms of phase, while the gain margin (GM) measures the system’s ability to withstand changes in system gain before becoming unstable.
Step 1: Understanding the Nyquist Plot.
In the given Nyquist plot, the points \( P \) and \( Q \) correspond to specific conditions on the open-loop frequency response of the system. The phase margin (PM) is the phase at the frequency where the open-loop transfer function crosses the unit circle, and the gain margin (GM) is the inverse of the magnitude at this frequency.
Step 2: Phase Margin Calculation.
Phase margin (PM) is calculated as the phase angle between the point \( P \) and the negative real axis, at the frequency \( \omega \) where the system crosses the unit circle. From the plot and the given data, the phase margin comes out to be \( 39.7^\circ \).
Step 3: Gain Margin Calculation.
Gain margin (GM) is calculated from the magnitude of the open-loop transfer function at the frequency where it crosses the negative real axis. From the plot and the data provided, the gain margin is \( 4.76 \).
Step 4: Conclusion.
Thus, the correct values for the phase margin and gain margin are PM = \( 39.7^\circ \) and GM = 4.76, which corresponds to option (A).
