Question

The Nyquist plot of a system is given in the figure below. Let \( \omega_P, \omega_Q, \omega_R, \) and \( \omega_S \) be the positive frequencies at the points \( P, Q, R, \) and \( S \), respectively. Which one of the following statements is TRUE?

• A. \( \omega_S \) is the gain crossover frequency and \( \omega_P \) is the phase crossover frequency
• B. \( \omega_Q \) is the gain crossover frequency and \( \omega_R \) is the phase crossover frequency
• C. \( \omega_Q \) is the gain crossover frequency and \( \omega_S \) is the phase crossover frequency
• D. \( \omega_S \) is the gain crossover frequency and \( \omega_Q \) is the phase crossover frequency

Hint:

In a Nyquist plot:
- The gain crossover frequency occurs where the plot intersects the unit circle (magnitude = 1).
- The phase crossover frequency occurs where the plot intersects the negative real axis (phase = -180°).

Answer 1

The Correct Option is D

In control systems and signal processing, the Nyquist plot is a graphical representation of a system's open-loop transfer function. The key aspects of the Nyquist plot include determining the gain crossover frequency and the phase crossover frequency:

1. Gain Crossover Frequency:
The gain crossover frequency is defined as the frequency at which the magnitude of the open-loop transfer function is 1 (or 0 dB). This is where the Nyquist plot intersects the unit circle.
In the plot, this corresponds to the frequency at which the distance from the origin to the curve is exactly 1.
At this point, the system's gain is 1, which means it neither amplifies nor attenuates the input signal.

2. Phase Crossover Frequency:
The phase crossover frequency is defined as the frequency at which the phase of the open-loop transfer function is \( -180^\circ \).
In the Nyquist plot, this corresponds to the point where the curve intersects the negative real axis.

Now let's analyze the plot given in the problem:

Step 1: Identifying the points on the Nyquist plot:
From the diagram, we see the points \( P, Q, R, \) and \( S \) on the Nyquist plot corresponding to different frequencies: \( \omega_P, \omega_Q, \omega_R, \omega_S \).
Point \( S \) lies on the unit circle, which indicates that this is the gain crossover frequency. Therefore, \( \omega_S \) is the gain crossover frequency.
Point \( Q \) intersects the negative real axis, indicating that this is the phase crossover frequency. Therefore, \( \omega_Q \) is the phase crossover frequency.

Step 2: Matching the frequencies with the choices:
Based on the above analysis:
\( \omega_S \) corresponds to the gain crossover frequency (where the plot intersects the unit circle).
\( \omega_Q \) corresponds to the phase crossover frequency (where the plot intersects the negative real axis).
This corresponds to option (D), which states: \( \omega_S \) is the gain crossover frequency and \( \omega_Q \) is the phase crossover frequency.

Thus, the correct answer is (D).