Question

Which plot is used to determine stability by looking at the encirclements of the point (-1, j0)?

• A. Bode Plot
• B. Root Locus
• C. Nyquist Plot
• D. Nichols Chart

Hint:

The point \( (-1, j0) \) has a magnitude of 1 and a phase of \( -180^\circ \).
It is the boundary point where the closed-loop system characteristic equation \( 1 + G(s)H(s) = 0 \) is satisfied.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks to identify which graphical stability analysis tool relies on counting the encirclements of the critical point \( (-1, j0) \) in the complex plane to determine closed-loop stability.
This graphical technique is highly useful because it allows engineers to evaluate closed-loop stability using open-loop frequency response data.

Step 2: Key Formula or Approach:
The stability assessment is based on the Nyquist Stability Criterion, which is derived from Cauchy's Principle of Argument (Argument Theorem).
The criterion is mathematically defined as:
\[ N = P - Z \] Where:
\( N \) is the number of clockwise encirclements of the critical point \( (-1, j0) \) by the Nyquist plot.
\( P \) is the number of open-loop poles of \( G(s)H(s) \) in the right-half s-plane (unstable open-loop poles).
\( Z \) is the number of closed-loop poles in the right-half s-plane (unstable closed-loop poles).

Step 3: Detailed Explanation:
Let us analyze how this criterion is applied:

• Nyquist Plot Analysis:
- The open-loop frequency response \( G(j\omega)H(j\omega) \) is plotted in the polar complex plane as \( \omega \) varies from \( -\infty \) to \( +\infty \).
- The stability of the closed-loop system requires that there are no unstable closed-loop poles (\( Z = 0 \)).
- Substituting \( Z = 0 \) into the Nyquist equation gives:
\[ N = -P \] - This means that for a closed-loop system to be stable, the Nyquist plot must encircle the critical point \( (-1, j0) \) in the counter-clockwise direction exactly \( P \) times.
- If the open-loop system is already stable (\( P = 0 \)), then the Nyquist plot must not encircle the critical point \( (-1, j0) \) at all (\( N = 0 \)) for the closed-loop system to remain stable.

• Comparison with other plots:
- Bode plots (Option A) use logarithmic magnitude and phase plots against frequency to determine stability via gain and phase margins.
- Root Locus (Option B) plots pole trajectories in the s-plane as a function of loop gain \( K \).
- Nichols Chart (Option D) plots gain magnitude in dB against phase angle on a single grid.

Step 4: Final Answer:
The Nyquist Plot is the graphical tool used to determine stability based on the encirclements of the critical point \( (-1, j0) \).