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📘 Bode Plot – Advanced Worked Examples (GATE Stability Practice)

This section provides deeper numerical problems on Bode Plot with full step-by-step stability procedure.

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🔹 Problem 1 – Gain Crossover & Phase Margin

Given:

G(s) = 5 / [s(1+s)]

Step 1: Identify Poles

• One integrator → slope = -20 dB/decade
• One pole at 1 rad/sec

Step 2: Magnitude Expression

|G(jω)| = 5 / [ω √(1+ω²)]

Step 3: Gain Crossover Frequency (|G| = 1)

Solve:

5 / [ω √(1+ω²)] = 1

Approximate solution:

ωgc ≈ 2 rad/sec

Step 4: Phase at ωgc

Phase:

• Integrator → -90°
• Pole → -tan⁻¹(ω)

Phase ≈ -90° - tan⁻¹(2) = -90° - 63° = -153°

Step 5: Phase Margin

PM = 180° - 153° = 27°

Conclusion: System stable but small margin.

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🔹 Problem 2 – Gain Margin Calculation

Given:

G(s) = 20 / [(1+s)(1+2s)]

Step 1: Find Phase Crossover Frequency

Total phase:

Phase = -tan⁻¹(ω) - tan⁻¹(2ω)

Set phase = -180°. Approximate solution:

ωpc ≈ 5 rad/sec

Step 2: Magnitude at ωpc

|G(j5)| ≈ 0.5

Convert to dB:

20 log10(0.5) = -6 dB

Step 3: Gain Margin

GM = +6 dB

Conclusion: Stable system.

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🔹 Problem 3 – Unstable Case

Given:

G(s) = 50 / [s(1+s)(1+0.1s)]

Step 1: High Gain

Large K shifts magnitude upward.

Step 2: At gain crossover

Phase becomes:

≈ -200°

Step 3: Phase Margin

PM = 180° - 200° = -20°

Conclusion: Unstable system.

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🔹 Problem 4 – Effect of Increasing Gain

If K increases:

• Magnitude shifts upward
• Gain crossover frequency increases
• Phase margin reduces
• System becomes less stable

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🎯 Final Stability Checklist

1. Find ωgc (0 dB point)
2. Calculate phase at ωgc → PM
3. Find ωpc (-180° point)
4. Check magnitude at ωpc → GM
5. PM > 0 & GM > 0 → Stable
6. PM < 0 → Unstable

Positive Phase Margin = Safer System