Nyquist and Nichols Stability Criteria Quiz Quiz

Explore essential principles of the Nyquist and Nichols stability criteria for control systems with this insightful quiz. Assess your understanding of frequency response analysis, phase and gain margins, and interpretation of stability plots in feedback systems.

1. Nyquist Plot and Encirclements

   Which statement correctly describes how the Nyquist plot determines system stability for a closed-loop control system with open-loop poles in the left-half plane?

   1. The number of clockwise encirclements of the point (–1,0) equals the number of right-half plane poles.
   2. The plot must not cross the origin for stability.
   3. Any counterclockwise encirclements of the point (0,1) indicate instability.
   4. Stability is indicated when the number of encirclements equals the total number of open-loop zeros.

   Explanation: According to the Nyquist stability criterion, the number of clockwise encirclements of the (–1,0) point in the complex plane must equal the number of open-loop poles in the right half of the s-plane for closed-loop stability. Option B is incorrect because the point (0,1) is not relevant to the Nyquist criterion. Option C is misleading since crossing the origin does not directly determine stability. Option D incorrectly relates encirclements to zeros instead of poles, which is not the basis for Nyquist analysis.

2. Phase Margin Interpretation

   If a closed-loop system's Nichols plot shows a phase margin of 45 degrees, what does this most directly indicate about its stability?

   1. The gain margin is exactly zero dB.
   2. The system will always have perfect tracking.
   3. The system is likely stable with reasonable damping.
   4. The system is unstable and may oscillate.

   Explanation: A phase margin of 45 degrees generally suggests that the system is stable and exhibits moderate to good transient response without excessive oscillations. Option B is false because a positive phase margin indicates stability, not instability. Option C confuses phase margin with gain margin, so it is irrelevant. Option D is incorrect since phase margin does not guarantee perfect tracking; it relates to stability and transient behavior.

3. Gain Crossover Frequency

   At which frequency is the phase margin measured in the context of the Nyquist or Nichols stability criterion for a feedback system?

   1. At the frequency where the imaginary part crosses zero
   2. At the phase crossover frequency where the phase is zero degrees
   3. At the frequency corresponding to the system's natural frequency
   4. At the gain crossover frequency where the magnitude is unity (0 dB)

   Explanation: Phase margin is defined at the gain crossover frequency, which is the frequency at which the open-loop gain magnitude equals unity (0 dB). Option B refers to phase crossover frequency, which is associated with gain margin, not phase margin. Option C is not used in Nyquist or Nichols analysis. Option D is unrelated because the natural frequency pertains to the time-domain response rather than frequency response margins.

4. Nichols Chart Axes Significance

   What do the axes of a Nichols chart represent when analyzing the stability of a linear control system?

   1. Real part on the x-axis and imaginary part on the y-axis
   2. Magnitude (absolute) on the x-axis and gain (dB) on the y-axis
   3. Frequency (Hz) on the x-axis and phase (degrees) on the y-axis
   4. Gain (in dB) on the y-axis and phase (degrees) on the x-axis

   Explanation: A Nichols chart displays the system's open-loop frequency response with phase (in degrees) on the x-axis and gain (in decibels) on the y-axis. Option B incorrectly lists frequency as an axis. Option C describes a Nyquist plot, not a Nichols chart. Option D does not match the typical format of a Nichols chart and is therefore incorrect.

5. Nyquist Criterion and Open-Loop Stability

   If a system with all open-loop poles in the left half of the s-plane has a Nyquist plot that does not encircle the point (–1,0), what can you conclude about the closed-loop system?

   1. The closed-loop system is stable.
   2. There is one right-half plane zero causing instability.
   3. The open-loop system must have a negative gain margin.
   4. The closed-loop system is unstable.

   Explanation: With all open-loop poles in the left-half plane and no encirclement of the (–1,0) point, the Nyquist criterion indicates that the closed-loop system is stable. Option B is incorrect because no encirclement and no right-half-plane poles means stability. Option C introduces zeros, which are not the focus of the Nyquist stability assessment. Option D implies a conclusion about gain margin not directly justified by the information given.