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📘 Nyquist Stability – Detailed Worked Examples (Clear Understanding)

This section explains Nyquist Stability very slowly and clearly. Each example shows how to calculate P, N and Z step by step.

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🔹 Example 1 – No RHP Pole (Simple Case)

Given:

G(s)H(s) = K / [s(1+s)]

Step 1: Count Open-Loop RHP Poles (P)

• Poles at s = 0 and s = -1
• No pole in Right Half Plane
• P = 0

Step 2: Nyquist Encirclement (N)

If Nyquist plot does NOT encircle (-1,0):

N = 0

Step 3: Closed-Loop RHP Poles (Z)

Z = N + P Z = 0 + 0 Z = 0

Conclusion: Stable system.

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🔹 Example 2 – One RHP Pole

Given:

G(s)H(s) = K / [s(s-2)]

Step 1: Count RHP Poles

• One pole at s = 2 (Right Half Plane)
• P = 1

Step 2: Stability Requirement

For closed-loop stability:

Z = 0

From formula:

Z = N + P

So:

0 = N + 1 N = -1

Meaning: Nyquist plot must encircle (-1,0) once in clockwise direction. If not → Unstable.

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🔹 Example 3 – Two RHP Poles

Given:

G(s)H(s) = K / [(s-1)(s-3)]

Step 1: Count RHP Poles

• Poles at s = 1 and s = 3
• P = 2

Step 2: Stability Condition

For stability:

Z = 0 0 = N + 2 N = -2

Nyquist plot must encircle (-1,0) twice clockwise.

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🔹 Example 4 – Gain Effect

Given:

G(s)H(s) = K / [s(1+s)(1+2s)]

Step 1: P Calculation

All poles in left half plane → P = 0

Step 2: Small Gain

Nyquist plot stays away from (-1,0) → N = 0 Z = 0 → Stable

Step 3: Large Gain

Plot expands and may encircle (-1,0) If encirclement clockwise → N = -1 Z = -1 + 0 = -1 (Not physically valid → unstable) Conclusion: Increasing gain may destabilize system.

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🔹 Example 5 – Practical GATE Question Type

Given: Open-loop has 1 RHP pole. Nyquist plot encircles (-1,0) once anticlockwise.

Step 1: Identify P

P = 1

Step 2: Identify N

Anticlockwise encirclement → N = +1

Step 3: Calculate Z

Z = N + P Z = 1 + 1 Z = 2

Conclusion: Two RHP closed-loop poles → Unstable system.

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🎯 Final Nyquist Procedure (Very Important)

1. Count open-loop RHP poles → P
2. Check Nyquist encirclement of (-1,0) → N
3. Compute Z = N + P
4. If Z = 0 → Stable
5. If Z > 0 → Unstable

Nyquist = Counting Encirclements for Stability

📘 Nyquist Stability – Advanced Worked Examples (GATE Deep Practice)

These problems strengthen conceptual clarity of Nyquist Stability. Carefully follow P, N and Z calculation in each example.

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🔹 Example 6 – Imaginary Axis Pole Case

Given:

G(s)H(s) = K / [s(s+2)]

Step 1: Identify Poles

• Pole at s = 0 (Imaginary axis)
• Pole at s = -2
• No RHP pole → P = 0

Important:

Pole on imaginary axis requires modified Nyquist contour.

Step 2: Stability Requirement

Since P = 0:

Z = N

If Nyquist does not encircle (-1,0) → Stable If it touches → Marginally stable

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🔹 Example 7 – Gain Variation Problem

Given:

G(s)H(s) = K / [(s+1)(s+3)]

Step 1: Poles

• All poles in LHP
• P = 0

Step 2: Small K

Plot small, no encirclement → N = 0 Z = 0 → Stable

Step 3: Increase K

Plot expands. At critical gain, plot touches (-1,0) → Marginal stability.

Step 4: Very Large K

Plot encircles once anticlockwise → N = +1

Z = 1 + 0 = 1

System becomes unstable.

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🔹 Example 8 – Two RHP Poles with Mixed Encirclement

Given:

Open-loop has 2 RHP poles Nyquist encircles (-1,0) once clockwise

Step 1: Identify P

P = 2

Step 2: Identify N

Clockwise encirclement → N = -1

Step 3: Compute Z

Z = N + P Z = -1 + 2 Z = 1

Conclusion: One RHP closed-loop pole → Unstable system.

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🔹 Example 9 – Zero in Right Half Plane

Given:

G(s)H(s) = K (s-1) / [(s+2)(s+3)]

Step 1: Poles

All poles in LHP → P = 0

Step 2: RHP Zero Effect

Zero affects shape but NOT P.

Step 3: Encirclement

If no encirclement → N = 0

Z = 0 + 0 = 0

System stable. Note: RHP zero does NOT directly affect stability condition.

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🔹 Example 10 – Marginal Stability Case

Given: Nyquist plot passes exactly through (-1,0)

Meaning:

Closed-loop characteristic equation has imaginary axis root. System is:

• Marginally stable
• Sustained oscillations

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🔹 Example 11 – Quick Exam Trick

If:

• P = 1
• No encirclement

Z = 0 + 1 = 1

System unstable immediately. No need to draw full plot.

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🎯 Final Advanced Stability Summary

• P counts only RHP poles (not zeros)
• Clockwise encirclement → Negative N
• Anticlockwise → Positive N
• Z = N + P
• Z must be zero for stability

Nyquist = Count P → Count N → Calculate Z → Decide Stability