The Nyquist plot and step response/transfer function is given. Match List-I (Nyquist plot) with List-II (corresponding step response/transfer function).
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Use the Nyquist stability criterion to quickly categorize plots:
- \textbf{No encirclement of -1:} Stable (response decays, e.g., I).
- \textbf{Encirclement of -1:} Unstable (response grows, e.g., II).
- \textbf{Passes through -1:} Marginally stable (response has sustained oscillations, e.g., III).
Step 1: Analyze Plot A.
The Nyquist plot (A) starts at some point on the positive real axis, circles the origin, and ends at the origin. It does not encircle the -1 point. This indicates a stable system. The circling behavior suggests oscillations that decay. This matches the damped oscillatory step response in Plot (I).
Step 2: Analyze Plot C.
The Nyquist plot (C) encircles the -1 point. An encirclement of the -1 point indicates instability. An unstable system's step response will grow without bound, possibly oscillating as it grows. This matches the growing oscillatory response in Plot (II).
Step 3: Analyze Plot B.
The Nyquist plot (B) passes through the -1 point. This indicates marginal stability. The system will have poles on the jw-axis. A marginally stable system, when given a step input, will oscillate with a constant amplitude that does not decay. This matches the sustained oscillation in Plot (III).
Step 4: Analyze Plot D.
The Nyquist plot (D) is given with a transfer function (IV), \( \frac{1}{(s+2)^2} \). This is a critically damped or overdamped stable second-order system. Its Nyquist plot will start at \(1/4\) on the real axis for \(\omega=0\) and move towards the origin for \(\omega \to \infty\) without any encirclements. The plot (D) matches this description.
Step 5: Combine the matches.
A \(\to\) I, B \(\to\) III, C \(\to\) II, D \(\to\) IV. This matches option (B).
