Power Flow Diagram & Power Developed by Synchronous Motor

The phasor diagram of a synchronous motor is shown below. From the phasor diagram, let,

• V = Supply voltage/phase
• I_a = Armature current/phase
• R_a = Armature resistance/phase
• α = Load angle
• Φ = Power factor angle.

Input Power to Motor :

Motor input power per phase is V I_a Cos Φ. Now, the total input power for 3-Φ star-connected motor is,

P = √ 3 V_L I_L Cos Φ

P = 3 V_ph I_ph Cos Φ

Where,

• V_L and I_L are line values.
• V_ph and I_ph are phase values.

Power Developed by Motor :

The mechanical power developed / phase is,

= Back emf * Armature current * Cosine of the angle between E_b and I_a

= E_b I_a Cos (α - Φ) for lagging PF

= E_b I_a Cos (α + Φ) for leading PF

The copper loss in a synchronous motor takes place in the armature windings. Therefore,

Armature copper loss per phase = I_a² R_a

Total copper loss = 3 I_a² R_a

By subtracting the copper loss from the power input, we obtain the mechanical power developed by a synchronous motor as,

P_m = P - P_cu

For three-phase,

P_m = √3 I_L I_L Cos Φ – 3 I_a² R_a

Power Output of the Motor :

To obtain the power output we subtract the iron, friction, and excitation losses from the power developed. Therefore, net output power, P_out = P_m - iron, friction, and excitation losses. The above two stages can be shown diagrammatically called as Power Flow Diagram of a Synchronous Motor.

Net Power Developed by a Synchronous Motor :

The expression for power developed by the synchronous motor in terms of α, θ, V, E_b, and Z_s are as follows. Let,

• V = Supply voltage.
• E_b = Back emf per phase.
• α = Load angle.
• θ = Internal or Impedance angle = Tan^-1 (X_r per Z_s).
• I_a = Armature current per phase = E_r per Z_s.
• Z_s = R_a + J X_s = Synchronous impedance.

Mechanical power developed per phase is given by,

If R_a is neglected, then Z_s ≈ X_s and θ = 90°. substituting these values in the above equation.