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📘 Nyquist Stability Criterion – Complete Theory & Worked Examples

Nyquist Stability Criterion determines closed-loop stability using open-loop frequency response. It is extremely powerful for high-order systems.

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🔹 1. Basic Concept

Closed-loop characteristic equation:

1 + G(s)H(s) = 0

Nyquist examines how G(jω) encircles the point (-1 + j0).

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🔹 2. Important Terms

• P = Number of open-loop poles in Right Half Plane
• N = Number of clockwise encirclements of (-1,0)
• Z = Number of closed-loop poles in Right Half Plane

Relation:

N = Z − P

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🔹 3. Stability Condition

For closed-loop stability:

Z = 0

Thus:

N = -P

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🔹 4. Nyquist Plot Construction Steps

1. Write G(s)H(s)
2. Substitute s = jω
3. Find magnitude & phase
4. Plot real and imaginary parts
5. Check encirclement of (-1,0)
6. Apply N = Z − P
7. Conclude stability

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🔹 5. Worked Example 1

Given:

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

Step 1: Poles

• One pole at 0
• One pole at -1
• No RHP pole → P = 0

Step 2: Stability Condition

Since P = 0, for stability:

N = 0

If Nyquist plot does NOT encircle (-1,0) → Stable.

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🔹 6. Worked Example 2 (With RHP Pole)

Given:

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

Step 1: Poles

• One pole at s = 2 (RHP)
• P = 1

Step 2: Stability Condition

For stability:

Z = 0 N = -P = -1

Meaning: One clockwise encirclement of (-1,0) required for stability.

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🔹 7. Gain Margin & Nyquist

Distance from (-1,0) at phase crossover gives Gain Margin.

• If plot passes left of (-1,0) → Stable
• If plot encircles incorrectly → Unstable

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🔹 8. Common GATE Questions

• Number of encirclements
• Effect of increasing gain
• Check stability without full plotting
• Relation between Root Locus & Nyquist

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🎯 Stability Checklist

1. Count RHP open-loop poles → P
2. Check encirclement → N
3. Calculate Z = N + P
4. If Z = 0 → Stable
5. If Z > 0 → Unstable

Nyquist Directly Connects Frequency Response to Stability