Question:
Consider the stable closed-loop system shown in the figure. The magnitude and phase values of the frequency response of \( G(s) \) are given in the table. The value of the gain \( K_I (> 0) \) for a \( 50^\circ \) phase margin is (rounded off to 2 decimal places).
\begin{tabular}{|c|c|c|}
\hline
$\omega$ in rad/sec & Magnitude in dB & Phase in degrees
\hline
0.5 & -7 & -40
\hline
1.0 & -10 & -80
\hline
2.0 & -18 & -130
\hline
10.0 & -40 & -200
\hline
\end{tabular}
\includegraphics[width=0.5\linewidth]{59.png}
Consider the stable closed-loop system shown in the figure. The magnitude and phase values of the frequency response of \( G(s) \) are given in the table. The value of the gain \( K_I (> 0) \) for a \( 50^\circ \) phase margin is (rounded off to 2 decimal places).
\begin{tabular}{|c|c|c|} \hline $\omega$ in rad/sec & Magnitude in dB & Phase in degrees
\hline 0.5 & -7 & -40
\hline 1.0 & -10 & -80
\hline 2.0 & -18 & -130
\hline 10.0 & -40 & -200
\hline \end{tabular}
\includegraphics[width=0.5\linewidth]{59.png}
\begin{tabular}{|c|c|c|} \hline $\omega$ in rad/sec & Magnitude in dB & Phase in degrees
\hline 0.5 & -7 & -40
\hline 1.0 & -10 & -80
\hline 2.0 & -18 & -130
\hline 10.0 & -40 & -200
\hline \end{tabular}
\includegraphics[width=0.5\linewidth]{59.png}
Show Hint
For phase margin calculations, adjust the gain such that the phase lag matches the required value at the crossover frequency.
Updated On: Jan 23, 2025
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Solution and Explanation
Step 1: Determine the gain crossover frequency.
From the phase response table, identify the frequency at which the phase reaches \( -180^\circ + 50^\circ = -130^\circ \).
Step 2: Calculate the gain \( K_I \).
Using the magnitude response at this frequency, calculate \( K_I \) to achieve the required phase margin:
\[
K_I = 1.11 \, \text{to} \, 1.13.
\]
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