## 24 February 2018

### 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
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 + XL 2 ) ]
Inductive reactance XL = 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 + XL 2 ) ]  / [ | V |2  R ] [ ( R2 + ( 2πfL ) 2 ]2
[ dP / P ] = – 2 XL 2 / [ ( R2 + XL 2 ) ]  [ df / f ]
Where Sin2 F = XL 2 / [ ( R2 + XL 2 ) ]
Therefore
[ dP / P ] = { – 2 Sin2 F [ df / f ] }
If the load power factor Cos F = 0.6, Sin F =0.8
[ dP / P ] = { – 2 Sin2 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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