📘 Synchronous Generator – Power Angle Equation
Power delivered by synchronous generator depends on torque angle δ. This equation is very important for stability analysis.
🔹 1️⃣ Assumptions
- Stator resistance neglected
- Uniform air gap (cylindrical rotor)
- Constant terminal voltage
🔹 2️⃣ Phasor Equation
For generator:E = V + jX_s I
Where: E = Internal generated EMF V = Terminal voltage X_s = Synchronous reactance δ = Power angle🔹 3️⃣ Power Developed (Per Phase)
After derivation:P = (V E / X_s) sinδ
For 3-phase system:P = (3 V E / X_s) sinδ
Most important formula.🔹 4️⃣ Maximum Power
Maximum power occurs when:δ = 90°
Therefore:P_max = (3 V E / X_s)
Very important result.🔹 5️⃣ Stability Condition
Stable operation: 0° < δ < 90° If δ exceeds 90°: Generator loses synchronism.🔹 6️⃣ Example
Given: V = 230 V E = 250 V X_s = 5 Ω δ = 30° Find total power. Solution:P = (3 × 230 × 250 / 5) sin30°
Step 1: = (3 × 57500 / 5) × 0.5 = (3 × 11500) × 0.5 = 34500 × 0.5P = 17250 W = 17.25 kW
🔹 7️⃣ Important Observations
- Power proportional to sinδ
- Independent of speed
- Higher excitation increases E → increases power capability
- Higher X_s reduces power transfer
🎯 GATE Important Points
- Remember P = (3VE/X_s) sinδ
- Max power at 90°
- Stable region below 90°
- Used in power system stability analysis
Power Transfer Depends on Torque Angle δ
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