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 / phase  =  V I_a Cos φ

Total input power for 3-φ star connected motor,

P  =  √ 3 V_L I_L Cos φ 

    =  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,

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

=   E_b  I_a Cos ( α - φ )   for lagging p.f

=   E_b  I_a Cos ( α + φ )   for leading p.f

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

Therefore,

Armature copper loss / 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

The power developed in a synchronous motor as follows.

Motor Input Power, P

  1. Stator ( Armature ) copper loss P_cu

  2. Mechanical power developed, P_m

         a. Iron, friction and excitation losses

         b. Output power, P_out

Net Power Developed by a Synchronous Motor :

     The expression for power developed by an synchronous motor interns of  α, θ, V, E_b and Z_s is as follows :

Let

   V  =  Supply voltage

   E_b  =  Back emf / phase

   α  =  Load angle

   θ  =  Internal or Impedance angle  =  Tan^-1 ( X_r / Z_s )

   I_a  =  Armature current / phase  =  E_r / Z_s

   Z_s  =  R_a + J X_s  =  Synchronous impedance

Mechanical power developed / phase,

Armature resistance is neglected

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