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

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 / ( R^{2}+ X^{2}) ] + j [ | V |^{2}X / ( R^{2}+ X^{2}) ]
= P + j Q

Therefore

P = [ | V |

^{2}R / ( R^{2}+ X^{2}) ]
Q = [ |
V |

^{2}X / ( R^{2}+ X^{2}) ]
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 / ( R

^{2}+ X^{2}) ]
( dP / dV ) / P

= [ 2 | V |
R / ( R

^{2}+ X^{2}) ] / [ | V |^{2}R / ( R^{2}+ X^{2}) ]
= [ 2 | V |
R / ( R

^{2}+ X^{2}) ] / [ | V |^{2}R / ( R^{2}+ X^{2}) ]
= 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 / ( R^{2}+ X_{L}^{ 2}) ]
Inductive
reactance X

_{L}= 2πfL
P = [ |
V |

^{2}R / ( R^{2}+ ( 2πfL )^{ 2}]
dP / df =

[ ( R

^{2}+ ( 2πfL )^{ 2}][ 0 ] – [ | V |^{2}R ][ 0 + 2( 2πfL^{ })( 2πL) ] / [ ( R^{2}+ ( 2πfL )^{ 2 }]^{2}
[ dP / df ] / P =

– [ |
V |

^{2}R ][ 2( 2πfL^{ })( 2πL) ][ ( R^{2}+ X_{L}^{ 2}) ] / [ | V |^{2}R ] [ ( R^{2}+ ( 2πfL )^{ 2 }]^{2}
[ dP / P ] = – 2
X

_{L}^{ 2}/ [ ( R^{2}+ X_{L}^{ 2}) ] [ df / f ]
Where Sin

^{2 }F = X_{L}^{ 2}/ [ ( R^{2}+ X_{L}^{ 2}) ]
Therefore

[ dP / P ] = { –
2 Sin

^{2 }F [ df / f ] }
If the load
power factor Cos F = 0.6, Sin F
=0.8

[ dP / P ] = { –
2 Sin

^{2 }F [ 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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