📘 Critical Clearing Time (CCT) – Derivation & Numerical
Critical Clearing Time is the maximum time allowed to clear a fault so that the system remains stable.
🔹 1️⃣ Basic Idea
From swing equation:(2H/ωs) d²Î´/dt² = Pm − Pe
During 3-phase fault: Pe ≈ 0 So:(2H/ωs) d²Î´/dt² = Pm
Acceleration is constant.🔹 2️⃣ Simplified Acceleration Equation
Let: M = 2H/ωs Then: d²Î´/dt² = Pm / M This is constant acceleration motion. Using motion equation:δ = δ₀ + (1/2)(Pm/M)t²
At critical condition: δ = δc So:t_c = √[ 2M(δc − δ₀) / Pm ]
This is Critical Clearing Time formula.🔹 3️⃣ Numerical Example
Given: H = 5 MJ/MVA f = 50 Hz Pm = 0.8 pu δ₀ = 20° δc = 40° Step 1: Convert degrees to radians 20° = 0.349 rad 40° = 0.698 rad Step 2: Calculate ωs ωs = 2Ï€f = 314 rad/sec Step 3: Calculate M M = 2H/ωs = 2×5 / 314 = 10 / 314 = 0.0318 Step 4: Apply formula tc = √[ 2 × 0.0318 × (0.698 − 0.349) / 0.8 ] = √[ 0.0636 × 0.349 / 0.8 ] = √[ 0.0222 / 0.8 ] = √(0.02775)tc ≈ 0.166 sec
🎯 Interpretation
If fault cleared before 0.166 sec → Stable If cleared after → Unstable This is why fast circuit breakers are required.🔹 Important Observations
- Higher inertia H → higher CCT
- Higher mechanical power → lower CCT
- Faster protection → better stability
CCT = Maximum Safe Fault Duration
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