📘 Synchronous Motor – Power Equation & Torque Derivation
Power developed in synchronous motor depends on torque angle δ. This is one of the most important derivations in Electrical Machines.
🔹 1️⃣ Assumptions
- Stator resistance neglected
- Cylindrical rotor (non-salient)
- Constant terminal voltage
🔹 2️⃣ Phasor Relation
For motor:V = E + jX_s I
Where: V = Terminal voltage E = Back EMF X_s = Synchronous reactance δ = Torque angle🔹 3️⃣ Power Developed
Electrical power input:P = 3 V I cosφ
Using phasor geometry and neglecting R: After derivation:P = (3 V E / X_s) sinδ
This is the most important formula.🔹 4️⃣ Torque Equation
Mechanical torque:T = P / ω_s
Therefore:T ∝ sinδ
Maximum torque when: δ = 90° Called pull-out torque.🔹 5️⃣ Stability Condition
Stable operation: 0° < δ < 90° If δ exceeds 90° → Motor loses synchronism. Very important concept.🔹 6️⃣ Example
Given: V = 230 V (phase) E = 200 V X_s = 5 Ω δ = 30° Find power per phase. Solution:P = (V E / X_s) sinδ
Substitute: = (230 × 200 / 5) × sin30° = (46000 / 5) × 0.5 = 9200 × 0.5P = 4600 W (per phase)
Total 3-phase power: = 3 × 4600 = 13.8 kW🔹 7️⃣ Important Observations
- Power proportional to sinδ
- Maximum power independent of δ beyond 90°
- Increasing excitation increases E
- Higher X_s reduces power capability
🎯 GATE Important Points
- Remember P = (3VE/X_s) sinδ
- Torque ∝ sinδ
- Max torque at δ = 90°
- Stable region below 90°
Synchronous Power Depends on Torque Angle δ
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