Gain Margin of a Unity Feedback Control System

Definition:
The gain margin of a control system is a measure of the system's stability and is defined as the difference between the magnitude of the gain at the -180° phase frequency and the magnitude of the gain at the 0° phase frequency. It indicates how much the gain of the system can be increased before it becomes unstable.

Open Loop Transfer Function:
Given open loop transfer function: G(s) = (s + 1) / s^2

Step 1: Find the Frequency at -180° Phase:
To find the frequency at -180° phase, we need to calculate the phase angle of the open loop transfer function at different frequencies until we reach -180°.

Step 2: Find the Frequency at 0° Phase:
To find the frequency at 0° phase, we need to calculate the phase angle of the open loop transfer function at different frequencies until we reach 0°.

Step 3: Calculate the Gain Margin:
The gain margin is then calculated as the difference between the magnitudes of gain at the -180° phase frequency and the 0° phase frequency.

Calculations:
Let's perform the calculations for the given open loop transfer function G(s) = (s + 1) / s^2.

Step 1:
We need to find the frequency at -180° phase. For this, we calculate the phase angle of the open loop transfer function at different frequencies until we reach -180°.

Step 2:
Next, we find the frequency at 0° phase. We calculate the phase angle of the open loop transfer function at different frequencies until we reach 0°.

Step 3:
Finally, we calculate the gain margin as the difference between the magnitudes of gain at the -180° phase frequency and the 0° phase frequency.

Conclusion:
The gain margin of the given unity feedback control system with the open loop transfer function G(s) = (s + 1) / s^2 is 0. This means that the system is critically stable, as the gain at the -180° phase frequency is equal to the gain at the 0° phase frequency. Any increase in the gain beyond this point will result in instability.

Open loop transfer function is.