Effect of Voltage and Frequency on Active Power of Load

The effect of variation in voltage and frequency on active power of load is summarized as below. The complex load power is given by

        S = P + jQ

           = VI

           = Y | V |²

           Where Y is admittance of load

        S = | V |² Y

           = | V |² / ( R – jX )

           = | V |² ( R + jX ) / ( R – jX ) ( R + jX )

           = [ | V |²  R / ( R² + X² ) ] + j [ | V |²  X / ( R² + X² ) ]

           = P + j Q

Therefore

        P =  [ | V |²  R / ( R² + X² ) ]
       Q = [ | V |²  X / ( R² + X² ) ]

• Therefore we can say that the active power and reactive power is directly proportional to square of the supply voltage.

Effect of change in supply voltage on active power of load

dP / dV

       = [  2 | V | R / ( R² + X² ) ]

( dP / dV ) / P

      = [  2 | V | R / ( R² + X² ) ] / [ | V |²  R / ( R² + X² ) ]

      = [  2 | V | R / ( R² + X² ) ] / [ | V |²  R / ( R² + X² ) ]

      = 2 / | V |

Therefore

       dP / P = 2 d | V | / | V |

• We can say that change in voltage directly affect the change in real power of the load. If there is 10% change in supply voltage, 20% change in the real power of the load.

Effect of change in supply frequency on active power of load

The active power is given by

P = [ | V |²  R / ( R² + X_L ² ) ]

Inductive reactance X_L = 2πfL

P = [ | V |²  R / ( R² + ( 2πfL ) ² ]

dP / df  =

    [ ( R² + ( 2πfL ) ² ][ 0 ] – [ | V |²  R ][ 0 + 2( 2πfL )( 2πL) ] /

    [ ( R² + ( 2πfL ) ² ]²

[ dP / df ] / P =

    – [ | V |²  R ][ 2( 2πfL )( 2πL) ][ ( R² + X_L ² ) ]  / 

       [ | V |²  R ] [ ( R² + ( 2πfL ) ² ]²

[ dP / P ] = – 2 X_L ² / [ ( R² + X_L ² ) ]  [ df / f ]

Where Sin² Φ = X_L ² / [ ( R² + X_L ² ) ]  

Therefore

        [ dP / P ] = { – 2 Sin² Φ [ df / f ] }

If the load power factor Cos Φ = 0.6, Sin Φ = 0.8

       [ dP / P ] = { – 2 Sin² Φ [ df / f ] }

                      = { – 1.28 [ df / f ] }

It means that if there is ten percentage drop in frequency, there will – 12.8 load increase.

Effect of change in frequency on the Induction motor

• If the frequency drop / reduced, the induction motor draws more current from the supply source therefore the torque increases. 
• If the load on the motor remains constant, the speed will increase. 
• The torque is directly proportional to square of the supply voltage. 
• If the supply voltage increases, the torque also increases. 
• However the induction motor is a constant power device, the induction motor draw less current to compensate increase in the voltage. 
• Therefore the change in the voltage does not affect the power output of the induction motor to some extent.

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